Preheat the oven to 375°F (190°C). In a small bowl, combine the salt, black pepper, onion powder, garlic powder, and paprika. Rub the chicken breasts with 2 tablespoons of olive oil, then evenly coat them with the seasoning mixture. Place the seasoned chicken on a baking sheet or plate and set aside.

In a medium mixing bowl, combine the chopped broccoli, shredded cheddar cheese, sour cream, softened cream cheese, minced garlic, salt, and black pepper. Stir until the filling is smooth and evenly combined.

Using a sharp knife, cut a deep pocket lengthwise into each chicken breast, being careful not to cut all the way through. Fill each pocket generously with the broccoli-cheese mixture, allowing a small amount to rest on top if needed.

Heat the remaining 1 tablespoon of olive oil in an oven-safe skillet over medium-high heat. Add the stuffed chicken breasts and sear for approximately 2 minutes per side, or until lightly browned.

Transfer the skillet to the preheated oven, or move the chicken to a baking dish if the skillet is not oven-safe. Bake for 25 to 35 minutes, or until the chicken reaches an internal temperature of 165°F (74°C). Remove from the oven and allow the chicken to rest briefly before serving.