Preheat the oven to 375°F and position the rack in the center of the oven.
Carefully create a pocket in each chicken breast, ensuring not to slice completely through. Generously season all sides with salt and pepper.
Heat one tablespoon of olive oil in a large oven-safe skillet over medium heat. Sauté the finely chopped onion until softened and slightly golden, about 5-7 minutes.
Stir in the minced garlic, cooking for 30 seconds until aromatic. Add the baby spinach, stirring frequently until it wilts. Remove the skillet from heat but do not wash it.
In a mixing bowl, combine the softened Boursin cheese, mozzarella, Italian seasoning, crushed red pepper flakes, and the spinach mixture.
Evenly distribute the cheese filling among the chicken breasts. Secure the openings with two toothpicks per chicken breast to prevent excessive filling from escaping.
Return the skillet to medium-high heat and add the remaining olive oil. Once hot, sear the chicken breasts for 5 minutes on one side. Flip and cook for an additional 3 minutes.
Transfer the skillet to the oven and bake the chicken, uncovered, for 15 minutes or until the internal temperature reaches 165°F. Allow the chicken to rest for 5 minutes before serving.
Remove the toothpicks before plating, and spoon the flavorful pan juices over mashed potatoes or rice for a delectable finish.