Butterfly your chicken breasts - hold your hand on top of the chicken breast and slice into the side of the chicken breast, cutting from one side to the other, leaving the opposite side intact. Your goal is to create a pocket for your filling. If needed, pound chicken breasts to create even thickness. Entire breast (closed) should be about 1 inch in thickness.
Season chicken breasts inside and out with salt, pepper, paprika, and garlic powder.
In a bowl, combine chopped broccoli florets, cheddar cheese, and mayonnaise. Stuff broccoli mixture into chicken, using a toothpick to hold closed if needed.
Add olive oil to skillet and heat over medium heat. Sear chicken breasts on each side for 2-3 minutes per side, until browned.
Move skillet to oven to finish cooking for 12-15 minutes, or until juices run clear. (Extra thick chicken that hasn't been pounded to 1 inch thickness will take longer to cook.)
Remove from oven and allow to rest for 2-3 minutes before serving.
Sharp cheddar can be substituted with any other shredded cheese on hand.
Fresh broccoli can be substituted with frozen broccoli, but it needs to be thawed first.
If your skillet is not oven-safe, transfer chicken to an oven-safe baking dish before baking.