Preheat the oven to 350°F.

Combine the vegetable oil and 4 tablespoons of the butter in a large skillet over high heat. Once the butter and oil are bubbling, add the chicken breasts to the skillet four at a time. Sear on each side until the chicken is golden, about 2–3 minutes per side, then transfer them to a large baking dish. Repeat with the remaining breasts and set aside.

Melt the remaining 2 sticks of butter in the same skillet over medium heat. Add the onion and garlic and cook, stirring occasionally, until the onions are soft, 15 to 20 minutes. Add the mushrooms, stir, and raise the heat to high. Allow the mushrooms to cook until most of their liquid has evaporated, 15 to 20 minutes. Add the tomatoes and cook, shaking the pan, for 10 minutes, or until the tomatoes begin to break down. Add the vinegar, wine, salt, and pepper and bring to a boil. Reduce the heat and let simmer for 15 minutes.

Pour the mushroom mixture over the chicken and place the pan in the top half of the oven for 15 to 20 minutes, or until the chicken has just cooked through. Remove from the oven and serve each piece of chicken generously topped with mushroom mixture.