Slice 1-inch off the top of the artichoke. Remove and discard the small leaves around each base/stem and snip off the thorny leaf tips
In a large pot, bring 2 inches of water to a boil. Add 1 tablespoon salt.
Halve the lemon. Set aside one half, then squeeze the juice from the other half into the boiling water. Add the artichokes, cover, and let steam until the artichoke bottoms pierce easily with the tip of a paring knife, 25 to 35 minutes, depending on their size. Drain the artichokes.
Finish the artichokes by cutting them in half lengthwise, and then use the point of a small spoon to scoop out the choke and some of the soft inner leaves from around each heart. Brush the artichokes with the olive oil and liberally season with salt and pepper.
Preheat a grill or grill pan to medium-high heat. Place the artichokes on the hot grill, cut-side down, until lightly charred, 6 to 8 minutes, rotating them 45 degrees about halfway through. Flip and grill for 2 minutes on the other side, just to heat through.
Arrange the artichokes on a platter and generously season with salt and pepper and serve with the aioli on the side and garnish with lemon wedges.
Prepare a grill or grill pan to high heat. Transfer the artichokes to the grill flat side up and cook, uncovered, until browned and lightly charred on one side (about 5 minutes). Flip and continue to cook until grill marks appear, about 4 to 6 minutes more. Remove the artichokes from the grill, drizzle with the remaining 1 tablespoon of olive oil, and season with more salt and pepper to taste.