Heat 2 tablespoons of olive oil in a large skillet over medium-low heat. Add the finely chopped shallots and cook for 1-2 minutes until softened. Stir in the minced garlic and Italian seasoning, cooking for just 30 seconds until fragrant. Increase the heat to medium, add the sliced mushrooms along with 1/4 teaspoon salt and 1/4 teaspoon pepper, and cook for 2-3 minutes until the mushrooms release their moisture and begin to brown. Add the balsamic vinegar and soy sauce, cooking for 1 minute to combine the flavors. Finally, add the fresh spinach and cook for 1 minute until just wilted. Transfer the filling to a plate to cool while you prepare the chicken.
Preheat your oven to 375°F. While the filling cools, carefully butterfly each chicken breast by slicing horizontally into the thickest part without cutting all the way through, creating a pocket for stuffing. Combine 1 teaspoon salt and 1/2 teaspoon pepper in a small bowl, then season both the inside and outside of each chicken breast generously with this mixture. Once the mushroom filling has cooled slightly, divide it evenly among the four chicken breasts, spooning it into the pocket of each one. Top the filling with equal portions of the grated Parmesan and shredded mozzarella cheese. Fold the chicken closed and secure with 2-3 toothpicks per breast to keep the filling from escaping during cooking. I like to tuck the edges under slightly to create a more uniform shape—this helps them cook evenly.
Heat 2 tablespoons of olive oil in a large oven-safe skillet over medium heat. Once the oil is hot and shimmering, carefully place the stuffed chicken breasts in the pan and sear for 5 minutes on the first side without moving them—this creates a golden crust that locks in juices. Flip each breast and cook for 3 minutes on the second side. Transfer the entire skillet to the preheated 375°F oven and bake for 10-15 minutes, until the internal temperature of the thickest part of the chicken reaches 160-165°F when measured with a meat thermometer. Remove the skillet from the oven and carefully transfer the chicken to a warm plate, leaving the pan drippings in the skillet for the sauce.
Place the skillet with the chicken drippings over medium heat. Add the minced garlic (1 tablespoon) and cook for 30 seconds, stirring constantly to prevent burning and to deglaze any flavorful browned bits from the bottom of the pan. Pour in the heavy cream and stir gently for 2-3 minutes, allowing the sauce to thicken slightly and incorporate all the pan flavors. Season to taste with salt and pepper, then stir in the freshly chopped parsley. I find that letting the sauce simmer for just a minute after adding the cream gives it a silkier, more luxurious texture—resist the urge to rush this step.
Remove the toothpicks from each chicken breast and place the stuffed breasts on serving plates. Spoon the warm creamy sauce over and around the chicken, ensuring each portion gets a generous coating of the rich pan sauce. Serve immediately while the chicken and sauce are still warm.