In a large pot, bring 1 gallon of water and salt to a boil. In a separate bowl, place the remaining gallon of water along with 4 lemon halves. To clean artichokes, remove the first layer of leaves (about 5–6 leaves), then trim the bottom of the stem. Cut the artichoke in half and remove the thistles from the center with a spoon. Place the cleaned artichoke in the lemon water mixture, and repeat until all artichokes have been cleaned. Once all artichokes are clean, boil them for 15–20 minutes or until a paring knife can be inserted into the heart of the artichoke and removed easily. Remove the cooked artichokes from the water, drain well and toss in olive oil. Lay artichoke halves heart-side down on a hot grill and cook for about 8–10 minutes, or until they begin to char. Flip them over and cook for 2 more minutes, then remove from the grill. Place the cooked artichokes on a serving plate, squeeze lemons over the top, drizzle with 1 tbsp olive oil and top with Parmesan cheese.