Fill a large pot, with a tight fighting lid, with about 2" of water and place a steamer basket in the pot.
Heat grill or grill pan over medium heat. If you have a charcoal grill you are heating for other things you are cooking that is perfect and will give the artichokes a fabulous grilled taste.
Cut the tops of the artichokes off. Then with kitchen shears cut off the thorns on the the tips of the leaves. Cut the artichoke in half vertically.
Using a spoon, scoop out the fuzzy choke in the middle of each half. It might take a little muscle, but not too bad.
Place the cleaned artichokes on the steamer basket and cover. Bring to a boil and let steam for about 30 minutes. Be sure and check half way through the cooking time that you don't need more water in the bottom of the pot.
Once the artichokes are tender remove from pot. Drizzle olive oil on each half. Generously sprinkle them with salt and pepper.
Grill each half for about 3 minutes a side or until you see nice grill marks.
Serve with melted butter combined with the juice of one lemon.