Place a Dutch oven or large soup pot over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add mushrooms, peppers, onion and the two smashed garlic cloves, and cook until tender, 7-8 minutes. Season with red pepper flakes, salt and freshly ground black pepper to taste.
While vegetables cook, rip the bread into small pieces and soak them in the milk in a mixing bowl for about 5 minutes, until most of the milk has been absorbed.
Add the tomatoes and chicken stock to the soup pot with the veggies and turn up the heat to bring the stoup to a bubble.
While you're waiting for the stoup to bubble, make the meatballs: Squeeze out the excess milk from the bread and discard it. To the mixing bowl with the soaked bread, add the ground turkey, egg, grated cheese, parsley, the remaining grated garlic clove, salt and freshly ground black pepper. Mix everything with your hands to combine.
Form the meat into 2-inch meatballs. Once the stoup is bubbling, drop in the meatballs then cover the pot and reduce the heat to a simmer. Cook until the meatballs are cooked through, 10-12 minutes.
To serve, ladle the stoup into shallow bowls and garnish with basil and shaved Parmigiano-Reggiano. Pass some crusty whole grain bread at the table for mopping up the leftovers.