Instructions
Step 1
Set a wire rack inside a rimmed baking sheet, place it in the center of the oven, and preheat to 200°F. In a large skillet over low heat, toast 1 cup of the flour, stirring continuously, until deep golden, 15–20 minutes. Turn off the heat, transfer to a small bowl, and set aside. Wipe out the skillet and return it to the stove.
Step 2
In a pie plate, whisk together the remaining flour and 2 teaspoons each of salt and black pepper. To the empty skillet, add 4 tablespoons of the butter and turn the heat to medium-low. When the butter is melted and foamy, dredge the quail in the flour mixture, shaking off any excess. Working in batches to avoid crowding the skillet, add the quail and cover. Fry, turning once halfway through cooking (and adding more butter if the pan looks dry), until browned and crisp on both sides and the juices run clear, about 16 minutes total. Using tongs, transfer to the rack in the oven.
Step 3
To the skillet, add the remaining butter and turn the heat to medium. Scrape up any browned bits, then add the mushrooms and cook, stirring occasionally, until softened and deep golden, about 5 minutes. Add the thyme, then gradually sprinkle in ⅓ cup of the browned flour, stirring to coat. (Reserve the remaining browned flour for another use.) Stirring continuously, gradually add the chicken stock. Cook until reduced slightly, about 5 minutes. Add the cream, turn the heat to medium-low, and simmer, stirring occasionally, until smooth and beginning to thicken, 12–15 minutes more. (If the gravy is too thick, add a bit more stock.) Season to taste with salt, black pepper, and hot sauce.
Step 4
To serve, spoon a generous amount of gravy onto four plates, then top with the quail. Pass any remaining gravy at the table.