Beat eggs in a shallow dish with a pinch of salt and pepper until well combined. Set aside.
Lay out 4 slices of bread. Spread mustard on 2 slices and mayonnaise on the other 2 slices.
On the mustard-spread slices, layer ham and shredded cheese. Top with the mayo-spread slices to form 2 sandwiches. Press down gently.
Heat a large nonstick skillet or cast iron pan over medium heat. Add 1 Tbsp butter and let it melt, swirling to coat the pan.
Working with one sandwich at a time, dip both sides in the beaten egg mixture, coating evenly but not soaking.
Place egg-coated sandwich in the hot pan. Cook 2-3 minutes per side until golden brown and cheese is melted. Add more butter to the pan before cooking the second sandwich.
Remove from pan and let cool for 1 minute before slicing. Serve immediately, dusted with powdered sugar or with jam/honey on the side if desired.