1. In a large bowl, combine lemon juice and 2 quarts cold water. Slice off top halves of artichokes and discard; break off and discard outer leaves down to tender, pale green inner ones. Trim off and discard stem ends. With a small, sharp knife, pare off dark green fibrous skin from artichoke bottoms and remaining stems. Cut artichokes in half lengthwise; scoop out and discard fuzzy centers and any thorny petals. Cut each artichoke half in half again lengthwise and drop into lemon water.
2. Peel onions; cut in half lengthwise, then slice lengthwise into 1/4-inch-thick slivers. Rinse fennel; trim off and discard stalks and root ends. Reserve a few leafy sprigs for garnish if desired. Cut heads lengthwise into about 1/4-inch-thick matchsticks. Shell peas. Rinse lettuce well and trim off and discard core; cut leaves into 3-inch lengths.
3. Drain artichokes. In a 5- to 6-quart pan over medium-high heat, frequently stir artichokes, onions, and fennel in olive oil until onions are limp, 7 to 8 minutes. Add 5 cups water and bring to a boil over high heat; cover, reduce heat, and simmer, stirring occasionally, until artichokes are tender when pierced, 12 to 18 minutes.
4. Add peas; bring mixture to a boil over high heat and cook until peas are tender when pierced, 2 to 3 minutes. Stir in lettuce and cook until barely wilted, about 30 seconds. Add salt and pepper to taste. Ladle stew into bowls and garnish with reserved fennel sprigs. Sprinkle with cheese to taste.