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 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) on a meat thermometer, 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.
Notes
Filling: Makes about 1 ½ cups
Storing: Cool completely and store in an airtight 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.