Preheat the oven to 400°F.
Cut a horizontal slice in each of the chicken breasts to form a pocket for the filling. Take care to cut evenly and not all the way through. You can use your fingers to gently open up the pocket.
In a small bowl whisk together the garlic powder, onion powder, paprika, salt, and pepper. Use this mixture to season the chicken on both sides as well as into the pocket you just cut.
In a medium bowl mix together the broccoli florets, red pepper, cheddar cheese, and sour cream.
Divide the filling mixture evenly between the 4 chicken breasts. Spoon the filling mixture into the pockets of the chicken breasts. If the chicken breast is really full then you can secure the chicken breasts with a couple of toothpicks. This assures that all your filling stays inside the chicken while cooking.
In an oven-safe skillet, heat the butter and olive oil on the stovetop over medium heat.
Add the chicken breasts to the skillet and sear on each side for about 4 minutes or until golden brown.
Remove the skillet from the stovetop, cover with a lid or aluminum foil, and place it in the preheated oven for 20 minutes, until the chicken is cooked through and a meat thermometer reads 165°F.
Plate the chicken and pour the drippings from the pan over the top of the chicken. Be sure to remove the toothpicks prior to serving.