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