Peel away the outer layer from the artichoke stem. Cut the top quarter off of the artichokes, then halve lengthwise.
In a medium pot, add the artichokes and cover with water. Bring to a boil and simmer for 15-20 minutes, or until you can easily peel a leaf off of an artichoke. Cool until you can easily handle them, then use a small paring knife to remove (and discard) any prickly bits and the inner purple leaves.
Preheat your grill to 350F-400F. Over direct heat, place the artichokes cut side down and grill for 10-15 minutes, or until the artichokes are crispy around the edges and you see grill marks. Remove the artichokes from heat and cool slightly.
In a small bowl, mix the mayo, soy sauce and garlic. Serve alongside the artichokes and lemon wedges.