Sautee the onion slices on low heat in a frying pan with 2 Tbsp olive oil and 1 Tbsp balsamic vinegar until the onions are translucent, around 5 minutes.
Add in the sliced mushrooms and cook for another 5 minutes before adding in the minced garlic, soy sauce, and red chili flakes. As the mixture cooks, scrape the caramelized bits from the bottom of the pan and let all the flavours come together for a few minutes. Set aside
Butterfly the chicken breasts by placing one hand over the top of the chicken and use a very sharp knife to cut through the centre of the breast about 3/4 of the way through. You are aiming to open up (butterfly) the chicken, stuff it with veggies and cheese, and then close it back up. If you accidentally cut all the way through, no big deal
Coat the outside of the chicken breast in 2 Tbsp of olive oil and the Italian seasoning. Divide the veggie mixture into 4 portions and spoon it into the middle of each butterflied chicken breast, then fit in the cheese. Close the flap over to "put the chicken back together" and place on the baking sheet
Bake at 400F for 30 minutes. Remove from the oven and let the chicken rest for 5 minutes before serving