Start the grill if you are using a charcoal or wood grill up to medium high heat (350°-400°). Start propane grill after draining the artichokes from boiling water.
Fill a very large pot full of water, you will need enough to cover the artichokes. Add some salt and bring up to a boil. While you are waiting for the water, prepare your artichokes.
Wash artichokes well, rinsing under running water. Tear off any small, loose leaves at the bottom of the artichoke. Trim the ends of the stems and cut about 1 inch off the top of the artichokes. You can trim off the tips (the sharp points) of the artichoke leaves with scissors, if you like.
Place artichokes gently into the boiling water, bring water back up to a boil. Reduce heat and cover pot. Simmer for 25-30 minutes depending on the size of your artichokes.
Drain the water and place artichokes upside down on a clean towel or baking sheet to dry, pat dry with paper towels as needed.
If using a propane grill light it now.
Cut artichokes in half lengthwise. Remove the choke (the inedible spiky part of the artichoke) with a small spoon or melon baller, being careful not to scoop out the heart of the artichoke.
In a small bowl, stir together the olive oil, lemon juice, salt, and pepper. Brush the mixture on to the cut side of the artichokes.
Place artichokes cut side down onto the hot grill for about 10 minutes with lid closed. Flip the artichokes over and cook for 5 more minutes.
Serve with magical lemon aioli.