Heat oil in a Dutch oven or large pot over medium heat. Add chicken and onion; cook and stir until browned, 5 to 7 minutes. Add flour; cook and stir until mixture is coated, 2 to 3 minutes.
Add water, potatoes, carrot, soup base, and Herbes de Provence. Cover and simmer, 15 to 20 minutes. Add mushrooms and peppers; simmer until vegetables are tender, 10 to 15 minutes more.
