Adjust oven rack to upper-middle position and preheat broiler on high. Arrange the torn pieces of bread in a single layer on a rimmed baking sheet. Broil until browned, about 5 minutes, shaking the pan as necessary for even browning. Remove from oven and set aside.
Meanwhile, in a Dutch oven or large pot over medium-high heat, heat oil until shimmering. Add sausage and onion. Cook, stirring occasionally, until sausage and cooked through and onions are softened, about 5 to 7 minutes.
Stir in garlic, thyme, and red pepper flakes until fragrant. Stir in tomatoes and their juice, 1 can of beans, and wine. Bring to boil, reduce heat, and simmer uncovered until stew thickens and darkens, about 10 to 15 minutes.
Add remaining can of beans to pot. If stew looks too thick, add more wine or water, 1/4 cup at a time until stew reaches desired thickness. Stir until heated through, 3 to 5 minutes.
Season to taste with salt and pepper. Garnish individual portions with chunks of bread and a drizzle of olive oil.