Trim artichokes with scissors to remove leaves' sharp edges. Bring 1 to 2 inches of water to boil in a large pot. Add artichokes, and steam 30 minutes or until inside leaves and heart are tender. Drain and cool slightly.
Cut each artichoke in half. Snip around fuzzy thistle (choke), and remove with a spoon. Grill artichoke halves 8 to 10 minutes or until hot, basting with butter and sprinkling with salt and pepper. Combine pesto and mayonnaise in a small bowl; serve on the side.
*Note: You can purchase the Smoked Tomato Pesto at hitchingpostcom. To make a similar version, combine 1 teaspoon chopped chipotle pepper in adobo sauce and 1 tablespoon minced sun-dried tomatoes in oil.