Preheat your oven to 375° Fahrenheit. Slice off the tops of the heads of garlic to expose the cloves. Wrap them individually, cut side up, in pieces of foil with a splash of olive oil. Place wrapped garlic on a cookie sheet and roast on the middle oven rack for 45 minutes. The garlic cloves should be soft and slightly browned; do not allow them to burn because they will become bitter. When cool enough to handle, squeeze out the garlic pulp; I use a small paring knife to help dig out the soft cloves. Set aside.
Lower oven heat to 350°. Season quail with salt, including inside the body cavity. In an oven-proof pan or skillet, heat about 1 tablespoon of oil over medium-high heat and brown the quail on all sides. Transfer the quail onto a plate and set aside.
Take the pan off the heat and add the roasted garlic and dry sherry. Return the pan over medium heat and allow the sherry to completely evaporate, stirring often. Next, add chicken stock, heavy cream, a bay leaf, and thyme, and bring to a simmer. Do not boil.
In a small bowl, mix 2 teaspoons of cornstarch and 1 tablespoon of water into a smooth slurry, and stir into the sauce. Simmer until thickened, stirring frequently. Season sauce to taste, and slightly smash the garlic pieces with the back of a spoon. Return browned quail to the pan, breast side up, along with any juices on the plate. Cover the pan with foil, transfer to a 350° oven, and bake for about 10-15 minutes, or until quail is cooked through.
Thin out the garlic sauce with more chicken stock, if needed. Discard thyme stem and bay leaf. Serve quail and garlic sauce with rice, mashed potatoes, boiled potatoes, or crusty bread.