Place the garlic, onion, celery and red bell pepper in food processor and pulse to fine chop. Add turkey, potatoes, egg, salt, pepper, poultry seasoning, and parsley leaves then pulse to process and combine the mixture. Remove bowl from base and blade from the bowl. In a medium mixing bowl combine the turkey mixture with 3/4 cup bread crumbs.
Heat a small skillet over medium heat with 2 tablespoons butter and 1 tablespoon oil. When butter melts add the mushrooms and cook until dark and tender, 5 to 6 minutes. Season the mushrooms with salt and pepper and add rosemary to the pan. Sprinkle the flour over the mushrooms and cook a minute. Whisk in the stock, bring to a bubble and thicken, 5 minutes, over medium-low heat.
Preheat a nonstick skillet with 3 tablespoons extra-virgin olive oil over medium heat.
Use a conical or round ice cream scoop to form 8 croquettes. Combine cheese and remaining 1/4 cup of bread crumbs on a plate. Coat the croquettes in cheese and bread crumbs and saute in hot oil until deeply golden all over, 5 to 6 minutes for each batch. Remove and drain on paper towels. Serve the croquettes with rosemary gravy poured down over them.