1. Combine 1 tablespoon oil, garlic, thyme sprigs, and lemon rind in a large zip-top plastic bag. Add the steaks to bag; seal and marinate in refrigerator for 3 1/2 hours, turning occasionally. Remove bag from refrigerator, and let stand for 30 minutes. Remove the steaks from marinade, and discard marinade. Sprinkle with 1/2 teaspoon salt and 1/2 teaspoon black pepper.
2. Place potatoes and bay leaf in a saucepan; cover with water. Bring to a boil; reduce heat, and simmer for 15 minutes or until crisp-tender. Drain; discard bay leaf. Cool completely; cut into 24 (1/3-inch-thick) slices.
3. Preheat the grill to medium-high heat.
4. Combine 2 cups water and 1/4 cup juice in a large bowl. Trim about 2 inches from top of each artichoke. Cut each in half vertically. Remove fuzzy thistle from bottom with a spoon. Trim any remaining leaves and dark green layer from base. Place artichoke halves in lemon water. Combine 2 tablespoons oil, 2 tablespoons juice, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a large bowl; stir with a whisk. Thinly vertically slice artichokes. Add sliced artichokes, arugula, and thinly vertically sliced onion to bowl; toss gently to combine.
5. Place steak, potato slices, and lemons on grill rack coated with cooking spray. Grill steak 4 minutes on each side or until desired degree of doneness. Grill potatoes for 3 minutes on each side or until tender. Grill lemons for 2 minutes on each cut side. Let the steak stand for 5 minutes. Cut steak diagonally across grain into thin slices.
6. Place 4 potato slices on each of 6 plates. Sprinkle evenly with remaining 1/4 teaspoon salt, remaining 1/4 teaspoon black pepper, and thyme leaves. Place 1 cup arugula mixture on each plate, and top each serving with 4 teaspoons cheese. Arrange 3 ounces steak on each serving, and drizzle evenly with remaining 1 tablespoon olive oil and remaining 2 tablespoons lemon juice. Serve with grilled lemon.
Wine Match: Brazin Lodi Old Vine Zinfandel ($16) a big, juicy wine loaded with vanilla, dark fruit, spice, and acidity to balance the lemony tang of the Tuscan-Style New York Strip. --Julianna Grimes