Heat a skillet over low to medium-low heat. (A non-stick or cast iron skillet works well.)
In a small bowl, mix the butter and mayonnaise together until smooth. Spread half of the mixture onto a slice of bread, making sure to coat the entire surface evenly. Place it buttered-side down, into the skillet. Top the bread with cheese.
Cover the skillet with a lid and cook on low for a total of 2-4 minutes. After 1 minute of cooking, use a spatula to lift the bread and check the progress. Adjust the heat of the burner up or down as necessary to avoid burning or undercooking. Once the bottom of the bread is golden brown and the cheese has started to melt, remove the lid and proceed to the next step.
Spread the mixture onto the second slice of bread and place it buttered-side up on top of the other in the skillet. Use a spatula to press the slice down so that the two slices adhere to one another. Carefully flip the sandwich over and press down again.
Cook the sandwich, uncovered for 2-4 more minutes or until the bottom is golden brown and the cheese has fully melted.