Combine water and 2 tablespoons juice. Working with one artichoke at a time, bend back outer green leaves of artichokes, snapping at base, until reaching light green leaves. Trim about 1/4 inch from top of artichoke. Cut off stem of artichoke to within 1/4 inch of base; peel stem. Cut artichoke in half vertically. Place artichoke halves in lemon water. Repeat with remaining artichokes. Set aside.
Melt 1 1/2 teaspoons butter in a large nonstick skillet over medium-high heat. Add onions; cook for 3 minutes or until onions begin to brown, stirring occasionally. Remove onions from pan, and set aside.
Add remaining 1 1/2 teaspoons butter and oil to pan, stirring until butter melts. Add shallots, mushrooms, and garlic; sauté 1 1/2 minutes or until shallots are tender. Add onions and wine; cook for 3 minutes or until wine almost evaporates. Drain artichokes. Stir artichokes and peas into onion mixture; cook 3 minutes or until artichokes are tender. Remove from heat; stir in remaining 1 1/2 teaspoons juice, parsley, salt, and pepper. Sprinkle with cheese.