Start by preheating a non-stick skillet or griddle over medium heat.
While the skillet is heating up, spread mayonnaise on one side of each slice of bread. This will give the bread a golden, crispy texture when it's cooked.
Place one slice of cheddar cheese on the un-mayoed side of two slices of bread. Then, place the remaining two slices of bread, mayoed side down, on top of the cheese slices to make two sandwiches.
Melt 1 tablespoon of butter in the skillet or griddle. Once melted, add the sandwiches to the skillet or griddle.
Cook the sandwiches for 3-4 minutes on each side, or until the bread is golden brown and the cheese is melted.
Once the sandwiches are cooked, remove them from the skillet or griddle and let them cool for a minute or two before serving.
Cut the sandwiches in half and serve hot.