To steam broccoli: Chop the broccoli florets into small pieces. Place the whole broccoli in a microwave-safe bowl and about 2 tablespoons of water. add water. Cover with plastic wrap and microwave on high for 2-3 minutes. Alternatively, steam on the stovetop or boil in boiling water for 1 minute.
Use a sharp knife to cut each chicken breast horizontally to create a pocket. Be sure not to cut about 1/2 inch from the ends and sides. Sprinkle chicken breasts generously with Italian seasoning, garlic powder, paprika, salt and pepper.
In a large bowl, mix steam and chopped broccoli, bell pepper, garlic, cheeses and salt and pepper; set it aside. Spread the mixture evenly on each chicken breast. Secure with toothpick if necessary.
Preheat oven to 400F. Heat 1 tablespoon of oil in a large cast iron or ovenproof skillet over medium-high heat. Add the chicken breasts and cook for 2-3 minutes on each side or until golden. Cover with aluminum foil and transfer to the preheated oven for 15-20 minutes or until golden brown.