Step 1

Place a rack in middle of oven; preheat to 375°. Trim any fat around edges of chicken. Working one at a time, place chicken between sheets of parchment paper and pound with a mallet or rolling pin until about ¼" thick. Season chicken all over with salt and pepper.

Step 2

Arrange chicken smooth side down on a work surface. Cover one half of each breast with one slice of ham and one slice of cheese, leaving space around the edges. Fold breast in half over ham and cheese and press edges together to firmly seal.

Step 3

Combine breadcrumbs, parsley, 1 tsp. salt, and ½ tsp. pepper in a shallow bowl. Whisk egg and 1 Tbsp. water in another shallow bowl. Spread flour on a plate.

Step 4

Press both sides of chicken in flour, shaking off any excess, then dip both sides into egg mixture. Coat all over with breadcrumb mixture, patting with your fingers to adhere. Transfer to a plate.

Step 5

Heat oil in a large heavy skillet over medium-high until shimmering. Cook chicken, turning occasionally, until browned on all sides, 2–3 minutes per side. Transfer to a wire rack set in a rimmed baking sheet; season with salt. Bake chicken until an instant-read thermometer inserted into the thickest part registers 165°, about 5 minutes.