Preheat oven to 400F.
Place a medium-size sauce pot over medium-high heat, add EVOO. Add the onion, carrot and celery, and cook for about 3-4 minutes. Add the thyme and cook for another minute. Add the crushed tomatoes, turn heat down to low and cook for 10-15 minutes.
In a medium-size mixing bowl, combine meats with egg, garlic, bread crumbs, milk, grated cheese, parsley, red pepper flakes and season with salt and pepper. Form into mini meatballs.
Add the meatballs to the sauce and cook for 15-20 minutes or until cooked through. Thin the sauce with chicken stock as the sauce thickens, or as needed.
Cut about 1/4 inch off of each end of the bun and then cut in half so you 2 mini buns. Place a couple strips of cheese on each mini bun and then place a couple meatballs on that. Top with a little sauce and enjoy!