Preheat oven to 350 degrees. Cut chicken into legs, thighs, wings and breasts; cut breast pieces in half crosswise. Season generously with salt and pepper then dredge in flour, tapping off excess. In a wide, 5-qt braising pan, heat vegetable oil with 2 T olive oil. Add chicken and brown lightly all over. Do not crowd pieces; if pan will not fit all of the chicken, brown it in batches. Remove chicken, add onions to pan and cook, stirring frequently, for 4 min. Add garlic, cook 1 min. Pour in wine, bring to a boil, and cook until reduced by half, approx. 1 min. Add tomatoes, capers and oregano; lightly salt and pepper then bring to a boil. Tuck chicken into the sauce, adjust the heat to a gentle simmer and cover the pan. Cook for 20 min, stirring a few times.
Meanwhile, in a large skillet, heat remaining 2 T olive oil over medium-high heat. Add mushrooms and peppers and toss until peppers are wilted but still crunchy, approx. 6 min. Season vegetables with a pinch of salt.
Add peppers and mushrooms to chicken. Cover, transfer to 350 degree oven and cook until chicken reaches 165 degrees internal temp, approx. 10 min. If necessary, add a small amount of water to maintain the level of liquid as the chicken cooks. Serve with Creamy Parmesan Polenta.
Creamy Parmesan Polenta
Bring water to a boil. Add salt and sprinkle in polenta slowly, stirring constantly. Turn down heat to medium-low and continue stirring slowly 10-15 min until polenta is softened. Add cheese, butter and crème fraîche. Remove from heat and adjust salt, if needed. Serve immediately.