Prep the chicken. Place each breast between two sheets of plastic wrap and pound to about ¼-inch thickness. Season lightly with salt and pepper on both sides.
Assemble your breading station. Set up three shallow dishes: flour in one, beaten eggs in the second, panko in the third. Keep them close together so you can work quickly.
Bread each cutlet. Lay a piece of chicken flat. Top with two slices of ham folded slightly to fit, then two slices of cheese. Roll tightly from the long end and secure with toothpicks at both ends. Roll first in flour (shake off excess), then egg (let excess drip), then press firmly into panko until well-coated. Set aside for 5 minutes before cooking.
Heat your pan. In a large skillet, heat butter and oil over medium-high heat until the butter foams but doesn't brown.
Cook the chicken. Add cutlets in batches (don't crowd — two at most). Cook 3-4 minutes per side until golden brown and crispy. The internal temperature should reach 165°F when you remove them from the pan.
Rest and serve. Transfer to a paper towel-lined plate for 2 minutes, then remove toothpicks before serving with lemon wedges if desired.
