Preheat Oven - Set the oven rack to the lower-middle position and preheat to 375ºF (218ºC).
Prepare the Broccoli - Wash and thoroughly dry. Trim off long stem pieces and cut them into no bigger than 1-inch pieces.
Make the Filling - In a large bowl, combine broccoli, softened cream cheese, cheddar cheese, green onions, minced garlic, ½ teaspoon salt, ¼ teaspoon black pepper, garlic powder, and onion powder. Set aside, or cover and refrigerate if not stuffing the chicken right away.
Prepare the Chicken - On the thickest side of the chicken breast, cut about a 4 to 5-inch long slit. Continue to cut a 2 to 3-inch deep pocket, leaving about a ½-inch uncut area to keep the other sides connected. Make sure not to cut through the bottom. Repeat with the remaining pieces.
Season the Meat - In a small bowl, combine 1 teaspoon salt and ½ teaspoon black pepper. Evenly season each side of the chicken with the mixture.
Stuff the Chicken - Evenly add the filling into the pocket of each chicken breast, about 5 to 6 tablespoons, do not overfill. If available, close the opening with toothpicks to secure the filling inside.
Sear the Chicken - Heat a large oven-proof skillet over medium heat. Once the pan is hot, add 2 tablespoons of olive oil. Add the chicken, presentation-side down, and lightly press to make contact with the pan. Cook until golden-brown on the surface, about 5 minutes. Flip and cook for 3 minutes. If using smaller chicken breasts, check for doneness as baking may not be necessary.
Transfer to the Oven - Place the pan in the oven. Bake until the meat is no longer pink and the internal temperature reaches 160 to 165ºF (71 to 74ºC), about 10 to 15 minutes, depending on the thickness.
To Serve - Transfer to a serving dish and rest for 5 minutes. Remove the toothpicks and serve while still hot.
Filling: Makes about 1 ½ cups
Storing: Cool completely and store in an air-tight container for up to 5 days.
Reheating: Cover and reheat in the microwave on high heat in 30-second increments until hot.
Using Frozen Broccoli: Defrost and drain the florets. Cut larger florets into ½-inch pieces. You will need 2 cups.
Make Ahead: The filling can be made separately, 3 days in advance. The chicken can be stuffed, covered, and refrigerated for up to 3 days before cooking.
Baking-Only Directions: Place stuffed chicken on a greased foil-lined sheet pan. Bake at 425ºF (218ºC) for about 20 to 25 minutes, depending on the thickness.