Preheat oven to 400°F. Line a baking sheet with parchment paper and set aside.
Mix cheese, ham and Cheezy Ranch dressing in a bowl. Season to taste with salt and pepper. Cut a horizontal slit along the thin, long edge of a chicken breast, almost all the way through to the opposite side.
Open up the pocket and place a quarter of the filling mixture inside. Close the edges of the chicken breast over the filling, pressing together to seal. Repeat with the remaining chicken breasts and filling.
Place the egg in one dish and the breadcrumbs mixed with the Cheez-It® cracker crumbs and ranch seasoning mix in another. Holding each piece of chicken carefully, dip into the egg, then press into the breadcrumbs.
Heat oil in a large nonstick skillet set over medium-high heat. Add chicken breasts, and cook until browned on one side, about 2–3 minutes. Transfer the chicken, browned side up, to the prepared baking sheet.
Bake until the chicken is cooked through or until an instant-read thermometer registers 165°F, about 20 minutes. Serve immediately.