Cut 1 of the lemons into quarters; halve and juice the remaining lemon and set aside. Fill a large saucepan three-fourths full of water. Add the lemon quarters, garlic, the 1 Tbs. salt and the peppercorns to the saucepan. Have ready a bowl of ice water. Snap off the tough outer leaves of the artichokes to reveal the pale inner leaves. Using a serrated knife, trim off 1/2 inch of the spiky tips. Using a paring knife, cut off the stems 1/2 inch from the bottom and peel the remaining stems.
Working with 1 artichoke at a time, cut them in half lengthwise and add to the saucepan. Cut out a parchment paper circle with a center vent and fit it inside the pan. Bring the water to a boil over high heat. Reduce the heat to medium-low and simmer until the artichokes are tender, 8 to 10 minutes. Using a slotted spoon, transfer the artichokes to the ice water. Drain and let dry on paper towels. In a large bowl, stir together the olive oil, reserved lemon juice and oregano. Add the artichokes and stir to coat. Season with salt and black pepper. Let stand for 20 to 30 minutes. Prepare a medium-hot fire in a grill. Brush and oil the grill grate or a vegetable-grilling basket.
Arrange the artichokes on the grate or in the basket directly over medium-high heat. Grill, turning often, until lightly charred on all sides and crisp-tender, 6 to 8 minutes. Arrange the artichokes on a platter and serve hot off the grill with the Meyer lemon aioli on the side. Serves 6.