Place 2 of the bread slices onto a cutting board. Add 2 slices of Cooper cheese to each slice of bread. Divide the sliced ham into two piles and add equal portions of ham on top of the Cooper cheese. Top the ham with the remaining 2 slices of Cooper cheese per sandwich. Complete the sandwiches with the remaining slices of bread.
Place a thick-bottomed skillet on a medium-high stovetop burner. Spread butter on each top slice of bread. Place each sandwich into the skillet, buttered side down. While the sandwich is grilling, spread butter on the other slice of bread for each sandwich. Grill until the first side is browned, then flip and brown the other side. Grilling should take about 4 minutes.
Once the bread is golden brown and the cheese is nice and gooey, remove the sandwiches from the heat and dig in!