Combine 2 cups water and 1 tablespoon juice in a large bowl. Cut off stem of each artichoke to within 1 inch of base; peel stem. Remove bottom leaves and tough outer leaves, leaving tender heart and bottom. Cut each artichoke in half lengthwise; place in water mixture, tossing to coat.
Heat oil in a large nonstick skillet over medium heat. Add onion; cook 5 minutes or until tender, stirring frequently. Add remaining 2 cups water, peas, and carrot; bring to a boil. Drain artichokes; add artichokes and potatoes to the pan. Cover, reduce heat, and simmer 45 minutes or until artichokes are tender. Uncover; cook 3 minutes. Stir in dill, mint, salt, and pepper. Remove from heat; stir in remaining 2 tablespoons juice.