Step 1: Preheat the oven to 425°F.
Step 2: To prepare the chicken, trim the excess fat from inside of the chicken cavity.
Step 3: Season the cavity with the herbes de Provence, bay leaves, salt, and pepper.
Step 4: Squeeze lemon juice into the cavity and then insert the used lemon quarters.
Step 5: Rub butter over the skin and season with salt and pepper.
Step 6: Tie the ends of the drumsticks together with kitchen twine, and set the chicken in a roasting pan, on a rack if you have one.
Step 7: Roast the chicken for 15 minutes, then decrease the heat to 350°F.
Step 8: Roast for an additional 15 minutes, then add the carrot and onion to the pan.
Step 9: Continue roasting, basting occasionally, until the juices run clear when the thickest part of the thigh is pierced with a knife, an additional 30 to 45 minutes.
Step 10: Remove the chicken to a cutting board and tent loosely with aluminum foil to keep warm.
Step 11: Using a slotted spoon, remove the vegetables to a warm platter and tent loosely with aluminum foil to keep warm.
Step 12: To make the sauce, remove all but several tablespoons of the fat from the roasting pan and place the pan over medium heat.
Step 13: Add the shallots and saute, stirring frequently, until softened, about 2 minutes.
Step 14: Add the wine and cook until it is reduced by half, 3 to 5 minutes.
Step 15: Add the chicken stock and increase the heat to high, scraping the skillet with a wooden spoon to loosen the browned bits.
Step 16: Cook until the sauce is slightly reduced, about 5 minutes more.
Step 17: Carve the chicken and pour any accumulated chicken juices from the cutting board into the roasting pan.
Step 18: Decrease the heat to medium, and whisk in the butter.
Step 19: Taste and adjust for seasoning with salt and pepper.
Step 20: Serve the chicken with the sauce on the side.
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