If the cheese is simply brown due to the Maillard Reaction and not yet burnt, it's safe to eat. There are several recipes for different kinds of fried cheese: Saganaki, Queso Frito, etc. so if you enjoy this kind of toasted cheddar you may want to try some of them too!
However, if the cheese has progressed beyond browning and is actually charred or burnt, it's probably not a great idea to eat it. There are several studies that suggest a potential link between HCA's created in burned foods and cancer, but beyond any health concerns it's basically just a bitter carbon blob at that point with no real nutritional or flavor value.
One suggestion if you are struggling to remove the baked-on cheese from your baking sheet is to use a silicone baking mat. It makes cleanup much easier and avoids any potential damage to the baking sheet from scrubbing, especially if you are concerned about a nonstick surface.