Peel the carrots and onions. Cut the carrots, onions, celery, and potatoes into bite-sized pieces.
Place the chicken, carrots, onions, celery, and potatoes into a slow-cooker or large pot.
Also add garlic cloves, crushed tomatoes, 4 cups of water, extra virgin olive oil, basil, salt, and pepper.
If cooking with a slow cooker: place the lid on and turn the slow cooker on. Cook for 6 hours on high or 8 hours on low.If cooking on the stove: turn the heat on high and cook for 15 minutes until the stew begins to boil. Cover with a lid and reduce the heat to low. Simmer for 1½ hours.
While the chicken stew is cooking, make the croutons. Cut the bread into 1-inch cubes. Lay them out on a sheet pan. Drizzle 1/2 cup of extra virgin olive oil over the cubes and sprinkle them with salt. Also add chopped basil. Massage the olive oil into the bread. Toast the bread for 15 minutes in a 350°F oven.
When the chicken is fall-apart tender, gently stir the stew to separate the chicken meat from the bones and pull it apart into smaller pieces. Serve the stew hot in a bowl, topped with croutons and extra basil.