Trim the tops from the artichokes and tips of leaves.
Cut in half lengthwise and remove the choke. Using ¼ of the lemon, squeeze juice onto cut side to prevent browning. Bring a large pot of water to boil. Preheat an outdoor grill. Add artichokes to boiling water and cook until tender, about 15 minutes. Drain well. Squeeze the juice from the remaining lemon into a small bowl. Add the olive oil and garlic. Season with salt and pepper. Brush the artichokes with the garlic oil. Grill the artichokes and baste with oil, turning frequently, until the tips are slightly charred, about 8 minutes. Serve with remoulade sauce.
Whisk all the ingredients for the remoulade sauce together until combined. Makes about 2 ¼ cups.