Melt 1 Tbsp. butter with 2 Tbsp. oil in a large skillet over medium-high heat. Working in batches, sprinkle chicken with salt and pepper, dredge in flour in a large bowl. Cook in a single layer, turning once, until golden, 5 to 6 minutes total. Remove to a plate; cover. Repeat with remaining chicken.
Add remaining butter and oil to skillet. Saute mushrooms until browned, about 7 minutes. Transfer to a plate; cover. Reduce heat to medium. Add onion and garlic. Cook, stirring, until tender, about 5 minutes.
Turn heat to high. Add wine. Cook until reduced by about 1/2, about 2 minutes. Add chicken, mushrooms, broth, tomato sauce and herbs. Stir. Bring to a boil, reduce heat to medium-low, partially cover and simmer until thickened, 10 to 15 minutes.