The trick to crisping the cheese properly is to use a gentle, medium-low flame the entire time.
Melt 3/4 of the butter in a large non-stick skillet over medium-low flame, coating pan thoroughly
Place the bread side-by-side in the skillet
Cover one slice evenly with half of the grated cheese (about 1/4 cup)
Top the cheese with the other slice of bread, with the butter side up
Sprinkle the buttered top evenly with about 2 tablespoons of the remaining grated cheese
Melt the remaining butter in the pan alongside the sandwich
Flip the sandwich immediately so the cheese-covered top is now facing down in the butter
Sprinkle remaining cheese on top of sandwich
Cook sandwich gently until the cheese on bottom is crisp and golden, about 3-4 minutes
Flip sandwich and cook until cheese on the second side is golden-crisp, another 3-4 minutes
Eat and enjoy immediately!