Preheat the oven to 400° F.

Place the roasted garlic, chopped rosemary, and 1 1/2 tablespoons of the olive oil in a food processor and purée until very smooth. Season with salt and pepper to taste.

Season the chicken halves on both sides generously with salt and pepper. Place the potatoes and onions in a large, shallow roasting pan or baking dish and drizzle with 3 tablespoons of olive oil Toss to make sure the vegetables are nicely coated in oil, then season with salt and pepper. Add all but one of the rosemary sprigs to the pan.

Heat a large nonstick skillet over medium-high heat. When hot, add the remaining 1 1/2 tablespoons of oil. Add the chicken halves to the pan, skin side down, and cook until the skin is crispy and golden brown, 3 to 4 minutes. Turn the chicken halves over and cook on the second side until golden, 2 to 3 minutes longer. Carefully transfer the chicken halves to the baking dish, positioning the chicken over the potatoes and onions. Spoon 2 tablespoons of the garlic-rosemary purée over each piece of chicken and, using the back of a spoon, spread it evenly all over the chicken. Transfer to the oven and cook until the chicken is golden brown and a thermometer inserted into the thigh registers 165°F, 45 to 60 minutes. Remove from the oven and allow the chicken to rest 10 minutes before carving and serving. (Reserve skillet for making pan sauce.) Test the potatoes to make sure they are done; if not, transfer the chicken to a cutting board and return the potatoes to the oven until cooked through.

While the chicken is resting, pour off any accumulated fat from the skillet that was used to brown the chicken. Return it to the stovetop over medium-high heat and, when hot, deglaze the pan with the white wine. When the wine has nearly completely evaporated, add the remaining rosemary sprig and the chicken stock and continue to cook until the stock has reduced by half, 4 to 5 minutes. Whisk in 2 tablespoons of the remaining garlic-rosemary purée and then add the cubed butter little by little, shaking the pan to help the butter melt before adding addtional cubes. Do not allow the sauce to boil or it will separate. When all of the butter has been incorporated, taste the sauce and season with salt and pepper if necessary. Remove the pan from the heat and cover to keep warm.

When ready to serve, carve the chicken halves into 4 pieces each and serve each guest a mix of dark and white meat, along with the roasted potatoes and onions, drizzled with the pan sauce.

Note:  Any remaining garlic-rosemary purée will keep in the refrigerator for up to several weeks. It can be used to flavor mashed potatoes, soups, stews, or sauces, or spread onto crostini or sandwiches for an extra layer of flavor.