Coat a large, wide pot with olive oil and put over high heat. Season the chicken with salt. To taste, and brown in the hot oil, working in batches, if needed. Once all the chicken is brown, remove it to a plate and discard most of the excess fat.
If needed to coat the bottom of the pan, add a tablespoon of olive oil. Add the onions and crushed red pepper, season with salt, to taste, and sweat over low heat for 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, to taste, 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 the bay leaves. Bring the liquid to a boil, then reduce the heat 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 arrange it on a pretty serving platter. Taste the sauce for seasoning and adjust if needed. Skim the surface of the sauce if excessively greasy. Spoon the sauce over the chicken and serve. Tastes like chicken!