Coat a large, wide pot with olive oil and bring to a high heat. Season the chicken with salt and brown in the hot oil, working in batches if needed. Once all the chicken is brown, remove to a plate and discard most of the excess fat.
If needed to coat the bottom of the pan, add 1 tablespoon olive oil. Add the onions and crushed red pepper, then season with salt, and sweat over low heat, 8 to 10 minutes. Add the garlic and sweat for 1 to 2 minutes. Add the peppers, and cook until they are soft, about 2 to 3 minutes. Add the mushrooms, season with salt and cook until they are soft. Add the white wine and reduce by half.
Return the chicken to the pan and add the tomatoes; taste for seasoning. Add the thyme bundle and bay leaves. Bring the liquid to a boil and reduce to a simmer. Partially cover the pot and simmer for 20 to 30 minutes. Check the level of the sauce periodically during the cooking process and replenish it with water as needed; the level of liquid should be about even with the chicken.
Remove the chicken from the pot and place it on a pretty serving platter. Taste the sauce for seasoning, adjust if needed. Skim the surface of the sauce if excessively greasy. Spoon the sauce over the chicken. Taste the chicken!