Whisk together yogurt, lemon juice, 3 tablespoons harissa paste, 1 1/2 tablespoons salt, thyme leaves, grated garlic, and 2 teaspoons pepper. Place chicken quarters in a large ziplock plastic bag. Pour yogurt mixture over chicken, and seal bag. Refrigerate at least 8 hours or up to 24 hours, turning bag occasionally.
Preheat oven to 375°F with oven rack in upper third of oven. Let chicken stand at room temperature 30 minutes. Scatter shallots, garlic, and lemon halves, cut sides down, in a heavy-duty roasting pan or 13- x 9-inch baking dish. Add thyme sprigs. Remove chicken from marinade, allowing excess marinade to drip off. Discard marinade. Sprinkle chicken with remaining 2 teaspoons salt and remaining 1 teaspoon pepper; arrange chicken, skin sides up, in a single layer in roasting pan. Add 1/4 cup water to pan.
Roast chicken in preheated oven until a meat thermometer inserted in thickest portion registers 140°F, 30 to 35 minutes. Turn broiler to high. Baste chicken with pan juices, and sprinkle chicken with paprika. Broil, basting chicken with pan juices every 2 to 3 minutes, until thermometer registers 165°F, 8 to 10 minutes.
Transfer chicken to a channeled cutting board; let rest 10 minutes. Meanwhile, squeeze 1 roasted lemon half and 6 roasted garlic cloves into pan juices; reserve remaining lemon half for serving and remaining garlic cloves for another use. Mash shallots and garlic with a fork. Pour pan juice mixture through a fine wire-mesh strainer into a bowl, pressing to extract liquid; discard solids. Whisk in remaining 2 teaspoons harissa paste. Pour accumulated juices from cutting board into harissa mixture in bowl, and stir. Pour pan juice mixture onto a serving platter, and top with chicken. Serve immediately with reserved lemon half.