Layer 1 slice of bread with 1 slice of cheese followed by both slices of ham and the last piece of cheese. Add the second piece of bread.
Use a piece of paper towel to rub an unheated heavy skillet with a little cooking oil. Ask a grown-up to help you place the skillet over medium heat.
As the skillet heats, whisk the egg, milk, and salt in a shallow bowl until frothy. Melt the butter in the pan.
As it melts, dip one side of the sandwich in the egg batter and then dip the other side.
Immediately place the sandwich in the pan. Grill the sandwich on the first side for about 3 minutes, until golden brown. Then flip it with a spatula and grill for second side for another 3 minutes.
Using a spatula, transfer the sandwich from the pan to a plate, slice it in half, and serve.