Remove thorns:  Using kitchen scissors, cut off the tips of the artichoke petals to remove the thorns.  Towards the top, you may find it useful to use a serrated bread knife to cut about 3/4 inches off the top.

Clean stem:  Pull off any small leaves from the stem.  Cut end of the stem and use a vegetable peeler to remove the tough fibrous outer layer of the stem, if necessary.  

Half the artichoke:  Cut the artichoke in half, lengthwise, starting from the stem.

Clean the choke:  Using a spoon, remove the hairy choke.  You don't want to ingest this, as it could get caught in your throat.  

Now you have a cleaned artichoke!

Fill up a large pot of water 2/3 full (anticipate the water level rising once you put the artichokes in).  Squeeze half of a lemon into the water and then put that half of lemon in the water.  You may add other optional aromatics, if you would like to infuse the cooking water with other flavors, like peppercorns and bay leaves.  Cover the pot and bring water to boil.  As the water is boiling, this is when I like to clean the artichokes.

Carefully place the artichokes in the boiling water.  If your artichokes refuse to stay in the water, you may want to put an upside down plate on top to weigh them down.  Just be careful that your water doesn't spill over!

Cook artichokes for 25-45 minutes, depending on the size of the artichoke.  You basically want to cook them until the outer petals can be easily pulled.

With a pair of tongs, pull the artichoke halves out and place them on a tray.  Drizzle some cooking oil on cut side and season with salt/pepper.  

Place cut side down on grill for a few minutes until desired grill marks are achieved.

Serve with Homemade Basil Pesto.  Enjoy!