Preheat the oven to 375°F (190°C).
Once the chicken has marinated (for 30 minutes at room temperature or longer/overnight in the fridge), fill the pocket with the mixture (one fourth of the mixture per chicken breast).
To a skillet over medium-high heat, add 1 tbsp of oil. Once the oil is hot (after about 1 minute), add the stuffed chicken and sear for about 2 minutes on each side (just enough to get some color on it).
Transfer it to a baking dish*, top with the remaining mozzarella, and bake in the preheated oven (middle shelf) until the chicken reaches 165°F/74°C (about 25 minutes, more or less depending on the size of the chicken breasts you have and the oven). *Note: If your skillet is ovenproof and large enough to hold all four chicken breasts, you can just use that instead of transferring the chicken to a baking dish. It will cook slightly faster as well (about 5 minutes faster as the skillet will already be hot).
Rest the chicken for at least 5 minutes before serving.