Fill a large pot with water and bring to a boil on the stove. Add a tablespoon each of olive oil and salt to the water.
Cut the top sharp ends off the artichokes, add them to the boiling water and cook for 30-40 minutes until the outside leaves pull off easily.
Remove the artichokes, drain and let cool. Cut them in half lengthwise and carefully remove the fuzzy center.
Mix the mayonnaise with the juice of half a lemon and set in the fridge.
Preheat the grill to 500°F and oil the grates with high-heat cooking oil.
Combine the melted butter, garlic, a pinch of salt and ground pepper to taste in a small bowl. Baste the inside of the artichoke halves with the butter mixture. Place the artichokes cut side down on the grill and cook until hot and you see grill marks.
Remove from the grill and serve with the mayo and extra lemon wedges. This dish pairs great with a glass of white wine and seafood on the grill!