Cut your chicken breasts into 8 pieces (I sliced my chicken in half vertically and then cut those pieces in half, giving me 8 relatively thin pieces). Place the chicken in a bowl with the milk and vinegar. Cover and let rest in the fridge for 10 minutes.
Meanwhile, place the flour on a plate and season it with salt and pepper to taste. Add the panko bread crumbs to another plate.
Working with one piece of chicken at a time, remove it from the milk, gently shake to remove excess liquid, and lightly coat it in the flour mixture. Dip the chicken lightly back into the milk and then move it to the panko bread crumbs to coat evenly by pressing gently on both sides. Continue this process until all the chicken is coated.
Preheat the oven to 375 degrees.
In a large, oven-safe skillet over medium heat, add the olive oil. Add the chicken to the pan and allow the bottom to form a golden crust (this will take about 5-7 minutes). Flip the chicken and allow the other side to become golden (about 3-5 minutes longer).
Fold each slice of deli ham and place one on top of each chicken piece. Top each piece with cheese. Move the pan to the oven and allow to bake until the cheese is melted and a thermometer inserted into the thickest part of the chicken registers 165 degrees.