Heat the oven to 220 C / 425 F with a rack in the center. Pat the chicken legs very dry with paper towels, including around the bone and under any loose skin.
Mix the softened butter, grated garlic, chopped thyme leaves, and 1/4 tsp salt in a small bowl until it forms a smooth paste.
Loosen the skin on each leg by sliding two fingers between skin and meat, starting at the thick end of the thigh and working toward the drumstick. Spoon roughly 2 tsp of the garlic butter into each pocket and press the skin back down to spread it. Keep the garlic off the outside of the skin, where it would burn.
Rub the skin of each leg with the olive oil, then season all over with 1 1/2 tsp salt, the pepper, and the paprika. Scatter the smashed garlic cloves and thyme sprigs on a 33 x 46 cm / 13 x 18 inch sheet pan and set the legs skin side up between them, at least 2 cm / 1 inch apart.
Roast for 40 to 45 minutes without flipping, until the skin is deep golden and an instant-read thermometer reads 175 F / 80 C in the thickest part of the thigh, not touching the bone.
Move the legs to a plate and rest for 10 minutes. Squeeze the lemon halves over the chicken, then spoon the pan juices over the top and serve with the soft roasted garlic cloves.