Rinse chicken and pat dry. Season the chicken pieces on all sides with salt and peppers. Dredge both sides of the chicken in flour. Just a light coating is fine.
In a large heavy sauté pan, heat butter and olive oil. Place the seasoned chicken pieces side to the pan. Cook until nicely browned, about 5 minutes, then turn over and lightly brown the other side. Remove chicken to a bowl, set aside.
Halve an onion from root to tip, and then cut it into slightly thick slices. Lop off the top and bottom of the pepper, pull out the innards, and cut it in half. Slice the pepper into thick slices. Next, slice up some mushrooms, not too thin and finely chop garlic.
Add more olive oil until you have about that much oil coating the pan. Add the bell pepper, onion and garlic to the same pan and sauté over medium heat until the onion is tender, about 5 minutes. Stir in salt and pepper. Add the mushrooms and cook, uncovered and stirring occasionally, until the mushrooms begin to brown and given up most of their moisture about 10 minutes. Add the garlic and cook, stirring, for 30 seconds.
Add the wine and scrape up any browned bits at the bottom of the pan. Simmer until the wine is reduced by half. Add the tomatoes with their juice, broth, capers and oregano. Test and taste to adjust seasoning. Simmer uncovered for another 5 minutes. Place the chicken pieces on top of the tomatoes and onions, skin side up. Lower the heat and cover the skillet with the lid slightly ajar. Continue simmering over medium-low heat until the chicken thighs is just cooked through about 30 minutes.
Using tongs, transfer the chicken to a platter and then vegetables. If necessary, boil the sauce until it thickens slightly, about 3 minutes. Spoon the sauce over the chicken, then sprinkle with the basil and serve with boiled rice or pasta.