Preheat the Oven - Set the oven rack to the lower-middle position. Preheat to 375ºF (191ºC).
Make the Filling - In a large bowl, combine softened cream cheese, mozzarella, parmesan, spinach, basil, garlic, ½ teaspoon salt, and ¼ teaspoon black pepper. Set aside, or cover and refrigerate if not using immediately.
Cut the Chicken - On the thickest side of the chicken breast, cut a 4 to 5-inch long slit. Continue to cut a 2 to 3-inch deep pocket, leaving about a 1-inch uncut area to keep the other sides connected. Make sure not to cut through the bottom. Repeat with the remaining pieces.
Season the Chicken - In a small bowl, combine 1 teaspoon salt and ½ teaspoon black pepper. Evenly season each side of the chicken with the mixture.
Fill the Pocket - Evenly spread the filling into the pocket of each chicken breast, about 3 to 4 tablespoons, do not overfill. If available, close the opening with toothpicks to secure the filling inside.
Brown the Surface - Heat a large oven-proof skillet or Dutch oven over medium heat. Once hot, add the 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 to 7 minutes. Flip and cook for 5 minutes. If using smaller chicken breasts, check for doneness as baking may not be necessary.
Bake the Chicken - Transfer the pan to 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. Rest for 5 minutes, remove the toothpicks, and serve hot.
Notes
Making the Filling Ahead: Store refrigerated in an airtight container for up to 3 days.