You want to use a mild cheddar cheese rather than a sharp one; mild cheddar has the perfect melt and flavor. A smear of mayonnaise on the outside of the bread will give your sandwich extra crunch.
The Sandwich Issue
Perfect Grilled Cheese
2 slices sandwich bread
2 ounces grated mild cheddar cheese
Heat a heavy pan over medium-low heat.
Thinly spread one side of each bread slice with butter. Slather the other side of each slice with mayonnaise and place the bread, mayonnaise-side-down, in the heated pan. Distribute the cheese evenly over the top of the buttered slices. Adjust the heat so the bread sizzles gently.
When the cheese is about halfway melted, use a flat metal spatula to flip one slice over on top of the other, and press lightly to encourage melting. Keep turning the sandwich, pressing gently, until it is crisp and golden on both sides, and the cheese is fully melted. Serve sliced on the diagonal.