Preheat broiler.
Place 1/4 cup of the EVOO in a small sauce pot, add the 2 crushed cloves of garlic. Place over low heat and gently cook for 3-4 minutes. Remove from the heat and reserve. Split open the French bread lengthwise, brush both cut sides liberally with the garlic oil and place under the broiler to toast until golden brown. Remove and chop the bread into extra-large, bite-sized pieces.
Cut the reserved poaching liquid veggies into large bite-sized pieces. In the pan with the remaining poaching liquid, add the crushed tomatoes and reserved chopped poaching liquid veggies. Bring the mixture up to a simmer. While the tomatoes are getting hot, chop the poached chicken into chunks. Once the tomatoes are simmering, add the chunked chicken, chopped tarragon and season with a little salt and pepper. Continue to simmer until the chicken is heated through.
Divide the garlic bread chunks between 4 serving bowls, top with stew and serve.