1. Combine the goose quarters, the giblets and the cold water in a heavy 8-quart casserole and bring to a boil over high heat, meanwhile skimming off and discarding the scum and foam as they rise to the surface.
2. Add the garlic, carrots, celery, onions, leek, the bouquet, thyme and salt, and reduce the heat to low. Partially cover the casserole and simmer the goose for about 1½ hours, or until the flesh of a thigh shows no resistance when pierced with the point of a small, sharp knife.
3. Then with tongs or a slotted spoon, remove the heads of garlic and set them aside. Transfer the goose quarters to a carving board. Cut away and discard the backbone and cut the goose into 8 serving pieces, trimming away any excess fat.
4. Strain the remaining contents of the casserole through a sieve set over a deep bowl and discard the vegetables. Let the stock settle for a few minutes; then, with a large spoon, skim as much fat as possible from the surface and reserve it in a separate bowl.
5. Peel the garlic, rub the cloves through a fine sieve with the back of a spoon and set the puree aside.
6. In a heavy 10- to 12-inch skillet, heat ¼ cup of the reserved goose fat until it is very hot but not smoking. Add the pieces of goose, a few at a time, and turn them frequently with tongs.
7. As they brown, arrange the pieces skin side up on a heated platter. Drape a sheet of foil loosely over the goose to keep it warm while you make the sauce.
8. In a heavy 1½- to 2-quart saucepan, heat 3 tablespoons of the remaining goose fat over moderate heat. Stir in the flour and mix together thoroughly.
9. Pour in 4 cups of the goose stock, add the pureed garlic and, stirring constantly with a whisk, cook over high heat until the sauce comes to a boil and thickens.
10. Reduce the heat to low and simmer uncovered for about 5 minutes to remove any taste of raw flour. Then combine the egg yolks and cream in a bowl and beat in about ½ cup of the simmering sauce.
11. Pour this back into the sauce in a thin stream, whisking constantly. Simmer for 1 to 2 minutes longer, then taste for seasoning, pour the sauce over the goose, and serve at once.
