Butter 1 side of each slice of bread.
In a pre-heated pan, place 1 slice of bread, buttered side down. Layer 2 slices of Cooper Sharp cheese on the bread. Sprinkle about 1/4 cup ham and 1/2 cup broccoli on cheese. Layer another 2 slices of Cooper Sharp cheese to cover the ham and broccoli. Top with another piece of bread, buttered side up.
Cook on medium heat for 2-3 minutes, then carefully flip and cook another 2-3 minutes on the other side. Check each side, cooking until bread is golden brown and cheese is melted. Repeat for remaining sandwiches.
Cut each sandwich in half and serve nice and warm.