Make the Mushroom Ragu
In a very large skillet, melt 1 tablespoon of the butter in 2 tablespoons of the oil. Add half of the mushrooms and season with salt and pepper. Cook over moderately high heat, stirring, until tender and just browned, about 7 minutes. Add half each of the shallots, garlic and thyme and cook until fragrant, about 3 minutes. Transfer to a bowl. In the skillet, melt the remaining 1 tablespoon of butter in the remaining 2 tablespoons of oil and repeat with the remaining mushrooms, shallots, garlic and thyme. Return all of the mushrooms to the skillet. Stir in the wine and cook until nearly evaporated, about 2 minutes. Add the chicken stock and season the ragù with salt and pepper; keep warm over low heat.