Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with aluminum foil.
Combine broccoli with 2 tablespoons water in a microwave-safe bowl. Microwave on high for 2 minutes. Drain.
Transfer broccoli to a large bowl. Stir in pepper Jack, mayonnaise, mushrooms, and garlic powder.
Season chicken breasts on both sides with paprika, salt, and pepper. Use a sharp knife to cut a slice through the middle of each breast to create a deep pocket, making sure not to cut all the way through. Evenly stuff each breast with broccoli mixture and place onto the prepared baking sheet.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, about 35 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).