2 (3–4-pound) chickens, each cut into 4 pieces, backbones removed
3/4 cup olive oil, divided
1/4 cup fresh lemon juice plus 1 lemon
12 rosemary sprigs, divided
10 garlic cloves, coarsely chopped
freshly ground black pepper
1 large pinch smoked paprika Tomato Jam
Arrange chicken in a glass baking dish. Drizzle with 1/2 cup oil and lemon juice. Coarsely chop leaves from 10 rosemary sprigs. Toss chopped rosemary and garlic with chicken to coat; season with salt, pepper, and paprika. Cover; chill for 3 hours or overnight.
Build a medium fire in a charcoal grill, or heat a gas grill to medium-high. Let chicken come to room temperature. Brush off marinade; grill chicken, turning occasionally, until browned and almost cooked through, about 20-22 minutes for legs and thighs, 16-18 minutes for breasts. Pour remaining 1/4 cup oil into a small bowl. Dip 2 rosemary sprigs in oil; occasionally baste chicken with sprigs until cooked through, about 5 more minutes.
Let chicken rest for 15 minutes. Transfer to a bowl. Cut lemon in half lengthwise, then cut thinly crosswise into half-moons. Add to chicken with any leftover basting oil; toss to coat. Serve chicken on a platter with Tomato Jam alongside.