About 1 hour before you plan to cook, remove the chicken from the refrigerator. Pat it very dry inside and out with paper towels. Season the cavity and the skin generously with salt and pepper. Rub the outside lightly with about 1–2 tbsp of the oil.

In a 3-quart nonreactive shallow saucepan, bring the wine to a rolling boil over medium-high heat. Once boiling, reduce to a strong simmer and cook for 8–12 minutes, or until the wine is reduced by about half.

Add the carrot, onion, celery, and garlic; maintain a low boil/simmer for about 10 minutes, or until the liquid has reduced by roughly half again. Add the demi-glace (and stock, if using) and simmer about 15 minutes, or until reduced by about half. Remove the pan from the heat, add the rosemary to steep, and set aside.

Preheat the oven to 425°F (220°C). Place the chicken breast-side up on a rack in a roasting pan or in a large ovenproof skillet. Roast until an instant-read thermometer inserted into the thickest part of the thigh (not touching bone) registers 165°F (74°C), about 60–80 minutes for a 4–5 lb bird.

Transfer the chicken to a cutting board, cover loosely with foil, and let rest 10–15 minutes before carving.

Strain the reserved sauce through a fine sieve into a bowl, pressing the vegetables with the back of a spoon to extract as much liquid as possible. Wipe out the saucepan, return the strained sauce to the pan, and bring to a gentle simmer. Whisk in the butter until melted. Remove from the heat and season with salt and pepper. (Makes about 1 cup)

Carve the chicken into 6 portions (2 breasts, 2 thighs, 2 drumsticks; or split breasts in half if you prefer more pieces). Arrange on plates, nap with the red wine demi-glace, and serve with Roasted Tomatoes and Summer Peas (or your preferred sides).