For the brine:
1 whole chicken (3½-4 lbs.), giblets discarded
For the garlic-rosemary paste:
For the sauce:
To prepare the brine, combine the salt, garlic and rosemary in a air-tight plastic bag, sealed with excess air pressed out. With a meat mallet or a rolling pin, pound the garlic cloves until smashed. Transfer the mixture to a large Dutch oven, bowl or stockpot. Stir in the hot water and let stand 10 minutes to release the flavors. Mix in the cold water and stir to dissolve the salt. Immerse the chicken in the brine, cover and refrigerate for 1 hour.
Remove the chicken from the brine and pat dry with paper towels. With an oven rack in lower-middle position, preheat the oven to 450˚ F. With a rack inside a roasting pan, spray the rack lightly with cooking spray.
To prepare the garlic rosemary paste, combine the rosemary, garlic, salt, pepper and 1 tablespoon of the oil in a small bowl. Mix well. Rub 1½ teaspoons of the paste inside the cavity of the chicken. Gently loosen the skin over the breast and thigh on each side of the chicken. Divide the remaining paste between the two sides of the chicken, slipping it underneath the skin. Rubbing over the surface of the skin, distribute the paste over the breast and thigh on each side. Tie the drumsticks together with kitchen twine. Brush the surface of the chicken with 2 teaspoons of the olive oil. Season with freshly ground pepper. Place the chicken breast-side down in the roasting pan and bake for 15 minutes.
While the chicken is cooking, toss the garlic cloves with the oil for the sauce. After the chicken has roasted for 15 minutes, toss the garlic cloves into the roasting pan and bake for 15 minutes more.
Remove the pan from the oven and lower the oven temperature to 375˚ F. Rotate the chicken so it is breast-side up. Brush the remaining 1 teaspoon of oil over the breast. Add 1 cup of the chicken broth and water to the roasting pan. Return the pan to the oven and continue roasting until the chicken is golden brown and an instant-read thermometer inserted in the thickest part of the breast reads 160˚ F. Transfer the chicken to a large serving platter and cover loosely with foil.
Transfer the roasted garlic cloves to a cutting board. Pour the liquid from the roasting pan into a liquid measuring cup; let sit a few minutes, then skim the fat off the surface. You should have about 2/3 cup of liquid (add water if needed). Peel the garlic and mash the cloves into a paste with a fork. Combine the roasting liquid in a small saucepan with the mashed garlic, the remaining ¾ cup broth, wine and rosemary. Heat over medium-high heat until simmering. Reduce heat and continue to simmer about 8 minutes, until partly reduced. Discard the rosemary sprig and adjust seasonings with salt and pepper to taste. Carve the chicken and serve with the sauce.