Adjust oven racks to lowest and middle positions and heat oven to 450°F. Pulse crackers and bread in food processor until coarsely ground. Drizzle in butter; pulse a few times to incorporate. Bake crumbs on rimmed baking sheet on middle rack, stirring occasionally, until light brown, 3 to 5 minutes. Transfer to shallow dish. Do not turn oven off.

Top each ham slice with 1/4 cup cheese and roll tightly; set aside. Pat chicken dry with paper towels. Using a paring knife, cut into thickest part of each breast to create deep pocket with opening of each 3 to 4 inches. Stuff each breast with 2 ham and cheese rolls and press closed. Season each breast with salt and pepper. Transfer chicken to plate, cover with plastic wrap, and refrigerate for at least 20 minutes.

Beat eggs and mustard in second shallow dish. One at a time , coat stuffed chicken lightly with flour, dip into egg mixture, dredge in crumbs, pressing to adhere. Transfer to clean rimmed baking sheet. Bake on lowest rack until bottom of breast is golden brown, about 10 minutes, and then move baking sheet to middle rack amd reduce oven temperature to 400°F. Bake until golden brown, 20 to 25 minutes. Transfer to platter and tent with foil and let rest for 5 minutes.