Position racks in the upper and lower thirds of the oven and preheat to 400 degrees . Line 2 large rimmed baking sheets with parchment. Place the chicken breasts on 1 sheet and the thighs on the other. Drizzle with the EVOO and squeeze the lemons on top. Sprinkle with the paprika and thyme; season with salt and pepper. Roast until the skin is crisp and the juices run clear when the pieces are pierced with the tip of a small, sharp knife, 45 to 50 minutes. Let cool for 15 minutes.
Remove the chicken skin, then pull and cut the meat off the bones. Tear the meat into shreds. (Save the bones, skin and any juices from the pans to make broth if desired; see bonus recipe at right.)