Preheat oven to 450 degrees F (230 degrees C). Pull the liver, gizzards, and other innards from the cavity. Dry the Whole Chicken (3.5 pound) inside and out with paper towels.
Remove the flaps of fat from the cavity opening--there should be one on each side. (If your butcher has removed the fat, you can use cold butter for this step.) Cut each piece of fat in half.
Work your fingers under the skin of each breast, going as far back as you can. Put two pieces of the cavity fat under the skin on each breast, one in back, toward the wing, the other further forward near the leg.
Poke the Lemon (1) with a poultry needle to make about 12 holes. Insert the lemon into the cavity. Now put the peeled Garlic (3 clove), Fresh Rosemary (3 sprig), and a little Salt (to taste) and Ground Black Pepper (to taste) in there and close up the cavity with the needle.
Rub the chicken all over with Butter (2 tablespoon) then sprinkle it with the Smoked Paprika (1/2 teaspoon), salt and pepper to taste.
Put the Butter (1 tablespoon) in a cast iron skillet and turn the burner to high. Once the butter has melted and is sizzling, put the chicken in the pan, breast side up. It should fit snugly, with the legs almost touching the wall of the pan.
Let the chicken sizzle in the pan for a minute or two, then put it in the oven.
Roast the chicken for 30 minutes, then turn off the oven and leave the bird in it for another 30 minutes. Do not open the oven.
Take the pan out of the oven and let it rest for at least 20 minutes. With an instant read thermometer, check the internal temperature at the thigh; it should be 165 degrees F (75 degrees C).
Transfer the chicken to a serving platter. Carve it and cover with aluminum foil to keep it warm while you heat up the pan juices. Let it boil for a couple of minutes, then turn off the heat, stir in Butter (1 tablespoon), whisking until it melts into the juices.
Serve the chicken pieces with the pan juices poured over.