1. In a food processor, crush stuffing mix to make crumbs. Use your hands to flatten chicken to an even thickness. Dip chicken into egg, then into stuffing crumbs.
2. In a frying pan, melt butter over medium heat. Add chicken and cook, turning carefully once or twice with a spatula, until crust is a rich golden brown and chicken is white throughout, 12 to 15 minutes. Remove to a platter and cover with foil to keep warm.
3. Add sage, broth, and half-and-half to pan drippings. Bring to a boil, stirring up browned bits clinging to bottom of pan. Boil, stirring often, until gravy is lightly thickened, 3 to 5 minutes. Season with salt and pepper to taste. Spoon gravy over chicken and serve.