Preheat oven to 350°F (175°C).
In a medium bowl, stir together 3/4 cup shredded sharp cheddar cheese, 2 ounces softened cream cheese, 2 tablespoons diced pimentos (drained), 2 sliced green onions, 2 slices cooked and crumbled bacon, a pinch of garlic powder, and a pinch of cayenne pepper until well combined. (This mixture can be made up to a day ahead—cover and refrigerate.)
Pat 4 boneless, skinless chicken breasts dry. Cut a horizontal pocket into the thick side of each breast, taking care not to cut all the way through.
Season the inside of each pocket and both sides of each chicken breast with salt and black pepper.
Evenly divide the pimento cheese mixture among the 4 chicken breasts and stuff each pocket. If needed, use toothpicks to close the openings.
In a shallow bowl, beat 1 large egg until combined. Place 1 cup dry breadcrumbs in a separate shallow dish.
Dip each stuffed chicken breast into the beaten egg, coating all surfaces, then press the chicken into the breadcrumbs to coat all sides, including the seams.
Heat 1 tablespoon butter and 1 tablespoon vegetable oil together in a large oven-proof skillet over medium heat until the butter is melted and the pan is hot.
Add the breaded chicken to the skillet and cook about 4 minutes per side, until the coating is golden brown.
Transfer the skillet to the preheated oven and bake for 12 to 14 minutes, until the chicken is cooked through (internal temperature 165°F) and the filling is heated.
Remove from the oven and let the chicken rest 3–5 minutes. Remove any toothpicks and serve.