Arrange the chicken pieces on a cutting board or rimmed baking sheet and sprinkle generously with salt and pepper. In a large skillet with a lid, heat olive oil over medium-high heat until just smoking.
Add the chicken skin-side down and cook until browned, about 5 to 6 minutes. Flip and continue cooking until browned on the second side, about 10 minutes total. Remove to a rimmed baking sheet or plate.
Pour off all but 2 tablespoons fat from the pan and return to shimmering. Add the onion, bell pepper, mushrooms, and garlic and sprinkle with salt to help coax out the water. Cook until the vegetables are tender, about 5 to 7 minutes.
Add the red wine, increase the heat to high, and bring to a boil. Cook uncovered until the wine is reduced by half, about 5 minutes. Stir in the tomatoes and their juice, basil, thyme, and red chili flakes, then add back the chicken.
Reduce heat to medium-low, cover, and simmer until all the chicken pieces reach an internal temperature of 165 degrees when tested at their thickest parts, about 25 minutes.
Remove the chicken to a serving platter or bowl and tent with aluminum foil. Bring to the sauce to a boil over high heat and cook until the sauce thickens, about 3 to 5 minutes. Season to taste with salt and pepper (I like 1/2 teaspoon salt and 1/4 teaspoon pepper). Pour over the chicken to serve.