Blanch the chopped broccoli in boiling water for 60-90 seconds until bright green. Immediately transfer to an ice bath to halt cooking. Drain thoroughly and pat dry with paper towels.
In a mixing bowl, combine the blanched broccoli, cheddar cheese, mozzarella cheese, cream cheese, minced garlic, red pepper flakes (if using), Italian seasoning, and onion powder. Mix thoroughly and set aside.
Place one hand flat on top of a chicken breast. Using a sharp knife, slice horizontally through the thickest part, stopping approximately ½-inch from the opposite side to create a 'hinge.' Open the chicken like a book.
Cover the butterflied chicken with plastic wrap and gently pound to an even ¼-inch thickness. Repeat with remaining breasts.
Season the inside of each butterflied chicken breast with salt, pepper, and garlic powder. Place approximately ¼ cup of the broccoli cheese mixture in the center, leaving a ½-inch border around the edges.
Fold the chicken closed over the filling. Secure with toothpicks or kitchen twine, spacing evenly to create a tight seal.
Mix remaining salt, pepper, garlic powder, and paprika in a small bowl. Brush chicken with olive oil and apply seasoning mixture evenly to all surfaces.
Preheat oven to 375°F (190°C). Heat an oven-safe skillet over medium-high heat. Add 1 tablespoon olive oil. Once hot, sear stuffed chicken breasts for 3-4 minutes per side until golden brown.
Transfer the skillet to the preheated oven and bake for 15-18 minutes, or until internal temperature reaches 165°F (74°C).
Remove from oven, tent loosely with foil, and allow to rest for 5 minutes before removing toothpicks/twine and serving.