Season chicken with 1 teaspoon salt and 1/2 teaspoon pepper. Dredge chicken in 8 tablespoons flour until evenly coated.
Heat olive oil and butter in a large cast iron skillet over medium heat. Working in batches, add chicken to the skillet in a single layer and cook until golden brown and cooked through, reaching an internal temperature of 165 degrees F, about 4-5 minutes per side; set aside.
Add garlic, shallot and mushrooms, and cook, stirring occasionally, until mushrooms are tender and browned, about 4 minutes; season with salt and pepper, to taste.
Whisk in remaining 1 tablespoon flour until lightly browned, about 1 minute.
Stir in chicken stock and wine, scraping any browned bits from the bottom of the skillet. Bring to a boil; reduce heat and simmer, stirring occasionally, until reduced and slightly thickened, about 3-5 minutes.
Stir in heavy cream; season with salt and pepper, to taste. Return chicken to the skillet.