In a medium bowl, whisk together eggs, milk salt and pepper. Set aside.
Assemble six sandwiches, each with one slice of ham and two slices of cheese between two slices of bread. Cut each sandwich into quarters.
Heat half of the melted butter in a large nonstick pan on medium heat, swirling to coat bottom of pan.
Dip half of the sandwich quarters into the egg and milk mixture, turning to coat. Immediately place in preheated pan, and cook for 2-3 minutes per side or until bread is golden brown and cheese is melting. Repeat with remaining butter and sandwiches. Serve hot.