Think of this as inside-out breaded chicken. Slideshow: More Chicken Breast Recipes
In a bowl, stir together the cornbread, egg, scallion and a pinch each of salt and pepper.
Place each chicken breast, smooth-side-up, on a work surface. Place the palm of one hand on top of one breast, then insert a pairing knife to make a length-wise pocket in each breast. Reserve 1/4 cup of the cornbread mixture, then divide the remaining cornbread mixture between the breasts, stuffing them full. Rub the chicken with the oil, then season all over with 3/4 teaspoon salt and 1/2 teaspoon pepper. Sprinkle the reserved 1/4 cup cornbread mixture over the chicken.
Preheat the oven to 425°F.
Place the chicken breasts on an oiled baking sheet and bake until cooked through, about 30 minutes. Let stand 5 minutes before slicing and serving.