The roles, he points out, are complementary.
"The job of a headteacher, in the loosest sense, is a spiritual one," Mr Moriarty says. "What you have to do is embody a certain set of values that are bigger than the school. Whatever you do is symbolic, and therefore spiritual.
"Training as a priest is all about reflecting on how systems are organised, on compassion, and on how we can create a community that is bigger than the sum of its parts and has a connection to something higher."
He adds: "In some ways, I think the same impulses that led me to explore priesthood also led me to explore school leadership. And working for a Jewish school, the crossover is huge.
"I am constantly struck by the deep conversations I have with Jewish colleagues and pupils. That has brought home to me the fact that, the deeper you go into spiritual life, the more common ground you find between different faiths."
Born and raised in Hampstead Garden Suburb and educated at Haberdashers' Boys' in Elstree, Mr Moriarty says he always "felt like I lived among and on the edges of the Jewish community".
He studied theology and philosophy at Oxford, before returning to London for his teacher training. Early teaching posts paved the way to a 12-year stint as head of religion and philosophy, and then as head of sixth form, at the UK's leading Haberdashers' Girls' school.
It had to take something special to lure him away.
"It's quite hard to leave a place like Habs," he explains. "But here was
something very different - a new school with a new vision.
"I had taught in a Jewish school before, so I knew it would have its act together. It would be well-organised and well-thought through, its governance would be sound, its finance would be sound, it would have sought excellent legal and professional advice.
"But above all, it was the vision of what the school was going to be: this pluralist, high-achieving environment. I just thought it was really exciting and worth leaving Habs for."
He knew he was boarding a ship sailing uncharted waters. As the first cross-communal Jewish secondary school in the UK, JCoSS, with its message of inclusivity, immediately distinguished itself from the existing Orthodox-run schools in the community. Suddenly, children who before may not have qualified to attend a Jewish secondary school would be able to pursue a Jewish education until the age of 18.
But, he says, he wasn't prepared for the full extent of "wariness" he and the school's founders initially faced.
"There was wariness because it was new," he says. "There was wariness because it was potentially providing over-capacity in Jewish schools and could also undermine the Jewish ethos of those other schools.
"There was an understandable objection to the notion of pluralism. I understand that there are good, principled reasons for saying pluralism doesn't work. I don't agree with them, but I respect them."
The wariness, he says, at times turned sour.
"We were the new kids on the block, which meant that people had to change the way they did things," he says. "To be honest, there was some hostility before we opened. That became suspicion and guardedness.
"But over time, it has evolved into understanding. The relationships between Jewish secondary school teachers are highly positive. We have a lot of useful collaboration."
He hopes that the United Synagogue will strengthen its ties with the school: "They are showing signs of warming up to us," he says. "I certainly hope this is the case, because more than a third of our students are from Orthodox backgrounds and what we earnestly want to do is provide them with role models from their own community.
"It has been admittedly difficult to get the Orthodox rabbinate over the threshold of the school. But we are working on it, and we hope that things will work out."
Mr Moriarty recognises the complaint among some sectors of the community that JCoSS's pluralist ethos "dilutes" the impact and influence of Jewish education. But he also wholeheartedly rejects the notion.
"I don't think we are leading the community astray," he says. "We now have a school that allows people to express their Jewish identity without compromise or pretence, who don't necessarily know how much they believe, but who care about their community and cultural heritage.
"That is the reality on the ground. I think we are being realistic."
He adds that the school has brought into the fold families who, without it, would have sought a secular education. In this way, he argues, "we have expanded the community's boundaries".
"I know that some people talk about 'marrying out' and others say 'marrying in'," he says. "I feel that we are a school that is 'marrying people in' from the fringes of the community.
"People are finding a home here that they may have not had before. I am proud of that, and I think the community should be too."
As any headteacher knows, juggling the sometimes dissident voices of the student body, parents, governorship and teaching staff can be a hard task, not least when you are based inside a community that values education - and arguing - above all else.
Has he ever been overwhelmed by the broiges?
"Not really," he says, "because the internal tensions of the community are almost identical to the internal tensions within the Church. That was kind of a home from home for me.
"And then there was the quality and character of the discussions taking place. I used to be the governor of a CofE primary school, where everyone was terribly polite and nobody was offended. The flip side was that things muddled along in ways that were not good."
He continues: "What struck me at JCoSS were the fantastically robust, animated arguments and discussions for the sake of heaven on matters that might be those of principle and policy, but might also be over how we could get the best deal on our IT contract.
"I was amazed by the passion, energy and determination to get things right - and to have a good old barney, or broiges, if a good old barney was required."
Five years down the line, the father of seven praises JCoSS's progress. He explains: "For five years, we have been saying that we are going to deliver amazing results. That we wanted to be judged by our values of turning children into mensches. Telling parents: 'Trust us, it will be okay.'
"So I am just over the moon by the fact that we have genuinely come through - not only with our results, but also our value-added scores. When I look at individual students, they are wonderful human beings.
"I see the mensches that we always hoped to create taking form before my eyes. Students who beforehand had almost no Jewish contact now talking about their sense of Jewish identity."
As for his aims looking forward, he remains committed to strengthening the school's ties with all sides of the community.
"I think that, as we forge deeper relationships with the Orthodox community, we can reassure their doubts and can then probably deepen the offer we give in terms of outreach to the spiritually and religiously committed.
"That doesn't mean turning moderately engaged people into highly observant families, but it means really getting them to articulate what their Jewishness means to them."
As for personal aims, he has a year and half to go before he is ordained, after which time he hopes to exercise his priesthood by exploring the "fertile ground" of interfaith work both inside and outside the school.
The hard hat may have come off, but he is still breaking new ground.
