In a shallow bowl mix together flour, bread crumbs, salt, ground red pepper and the zest from the second lime. In a second bowl mix together the egg and the juice of the second lime.
Preheat oven to 350 degrees. Wash and dry each chicken breast. Coat each breast in egg mixture and then coat in flour and bread crumb mixture. In a large oven safe skillet, melt butter over medium heat then add coated chicken and cook on both sides for about 10 minutes until golden brown. Drizzle the apricot sauce over chicken and bake in the oven for about 20 minutes. Serve over rice.