Turkey Vegetable Soup with Stuffing Dumplings

Name: Turkey Vegetable Soup with Stuffing Dumplings
Ingredients: Carcass from one 12 to 14 pound roasted turkey, picked clean; 2 large onions, one quartered and one chopped; 4 peeled carrots, 2 coarsely chopped and 2 sliced; 4 stalks celery, 2 coarsely chopped and 2 sliced; 6 garlic cloves, 4 smashed and 2 chopped; 1 bay leaf; 10 whole black peppercorns; 1 tablespoon extra-virgin olive oil; 2 large eggs; 6 tablespoons all-purpose flour, plus more as needed; 1/2 teaspoon salt, plus more as needed; Freshly ground black pepper, to taste; 2 cups leftover stuffing; 2 sprigs fresh thyme; 2 cups shredded leftover turkey meat; 1 cup leftover corn kernels