Slice tops off artichokes, pull off small leaves, trim stems, and snip off thorny tips. In a large pot, bring 1 to 2 inches of water to a boil. Add 1 tablespoon salt, the juice of 1 lemon, and artichokes; cover and steam until artichoke bottoms pierce easily, 20 to 40 minutes. Drain artichokes. When cool enough to handle, cut each in half lengthwise and scrape out fuzzy center. In a bowl, combine garlic, olive oil, 2 tablespoons lemon juice, 1/2 teaspoon salt, and pepper. Brush artichokes with garlic mixture and set, cut side down, on a grill over medium heat. Grill, turning once, until lightly browned, 8 to 11 minutes.
Green olive dip. In a blender, whirl 1/2 cup chopped parsley, 5 tablespoons extra-virgin olive oil, 2 tablespoons chopped green olives, 1 tablespoon drained capers, 1 tablespoon lemon juice, 1/2 teaspoon Dijon mustard, 1/4 teaspoon pepper, and 1/8 teaspoon salt until chunky.