Using a vegetable peeler or a medium knife, cut Parmesan into thin shavings. Set aside.
Squeeze juice from lemons into large pot; add lemons.
Fill pot with water. Cut stem and top 1/3 from 1 artichoke. Starting at base, bend leaves back and snap off where they break naturally; continue until all tough outer leaves have been removed. Using small sharp knife, trim outside of base until smooth and no dark green areas remain.
Cut artichoke lengthwise into quarters. Cut out choke from quarters. Cut each quarter in half, making 1/2-inch wedges. Place artichoke pieces in lemon water. Repeat with remaining artichokes. Cover pot and bring to boil. Boil artichokes until tender, about 10 minutes. Drain; set aside. (Can be prepared 1 day ahead. Cover and refrigerate.)
Cook 24 baby carrots in large pot of boiling water 5 minutes. Drain carrots well. (Carrots can be prepared 1 day ahead. Cover and refrigerate.)
Heat 1/4 cup olive oil in heavy large pot over medium-high heat. Add tomatoes and mushrooms. Sauté until tomatoes and mushrooms release juices, about 8 minutes. Add artichokes, carrots, baby peas and lima beans.
Cook until lima beans and peas are tender and juices thicken slightly, stirring often, about 10 minutes. Add 1/4 cup butter and stir until melted. Season vegetables to taste with salt and pepper. Transfer ragout to large bowl. Toss with balsamic vinegar and sprinkle with Parmesan shavings.
