3 pounds chicken backs and necks, cut into 2-inch pieces
¼ cup extra-virgin olive oil
12 whole garlic cloves, peeled
6 sprigs thyme
2 quarts chicken stock
3 tablespoons unsalted butter
1 shallot, peeled and finely chopped
½ cup brown chicken demi-glace
1 tablespoon sherry vinegar, plus more as needed
1 tablespoon finely chopped Italian parsley
For the Demi Glace:
Rinse the bones well under cold water and pat them dry with paper towels to remove excess moisture.
Heat a heavy pan or pot large enough to hold chicken bones over medium heat and add the oil. Once it starts to smoke, add chicken bones in a single layer and cook, in batches if needed, until golden brown, about 7 minutes. Turn the bones, adding the cloves of garlic and thyme, and cook the second side until golden brown, about 7 minutes more. Transfer the browned chicken bones to a medium bowl and repeat with remaining bones. Transfer all the remaining bones and juices to the stockpot.
Pour chicken stock over the top and keep pot over a low simmer. Let the chicken jus simmer for 2-3 hours. Make sure to skim any excess fat off of sauce.
Drain the liquid through a fine strainer and reserve, discarding the bone and solids.
Chill the broth in the freezer until the fat on the top sets. Scoop off the fat and discard it. Place remaining broth in a small pot and reduce to ½ cup. Set aside until ready to use.
For the Brown Butter Sauce:
Place 1 tablespoon of butter in a small saucepan over medium-low heat and sweat the shallots until translucent but not brown. Add the chicken demi-glace and let thicken to a sauce consistency. Keep warm.
Meanwhile, in another small saucepan, brown the remaining 2 tablespoons of butter over medium-low heat until the butter turns amber and smells nutty, and the milk solids fall to the bottom of the pan and brown. Stir the browned butter into the chicken sauce. Stir in the sherry vinegar and simmer, stirring often, until thicken slightly, about 5 minutes. Season with salt, pepper and more vinegar, as needed. Add the parsley just before serving.