Preheat the oven to 350°F (175°C).
Make the pimento cheese: in a medium bowl, combine the shredded cheddar, softened cream cheese, diced drained pimentos, sliced green onions, crumbled bacon, garlic powder, and cayenne; mix until smooth. Cover and refrigerate if making ahead.
Prepare the chicken: cut a slit along the thick side of each breast to form a pocket, being careful not to cut through. Season both sides and the inside of each breast with salt and black pepper.
Stuff each chicken breast with the pimento cheese mixture, dividing it evenly among the four breasts.
Set up dredging stations: beat the egg with 1 tablespoon of water in a shallow bowl, and place the breadcrumbs in another shallow dish. Dip each stuffed breast in the egg wash, coating both sides, then press into the breadcrumbs to coat.
Heat the butter and vegetable oil in a large oven-proof skillet over medium heat. When hot, add the breaded chicken and cook until golden, about 4 minutes per side.
Transfer the skillet to the preheated oven and bake until the chicken is cooked through, about 12 to 14 minutes. Remove from oven and let rest briefly before serving.