Can you eat a veggie burger with melted cheese? Probably yes, though there's reason to think maybe not. What could be wrong with it? After all there is no prohibition on eating milk with textured-soy protein. The possible problem comes from the halachic concept of marit ayin, literally "how it looks". This idea comes into play in cases where something would look bad if people don't understand what's really going on.
Based on this, Rabbi Moshe Isserles, the Rema, in 16th-century Poland forbade cooking meat in almond milk, which was apparently a local custom where he lived. The two reasons given for prohibitions based on marit ayin are either that someone watching may be misled by your example and come to do something forbidden, or that they might suspect you of wrongdoing and one should try to live above suspicion. Today, though, since veggie burgers are common, people are unlikely to get confused.