Preheat oven to 500° with rack set in upper third. Meanwhile, cut nopales into 1/2- by 2-in. strips. In a frying pan, sauté nopales over high heat with 2 tsp. oil until pan is mostly dry, 5 minutes. Reduce heat; cook until barely tender, 3 minutes more. Set aside.
In a bowl, whisk lemon juice, 1 tbsp. oil, 1/4 tsp. salt, and half of chipotle. Gently stir in avocado; set aside.
In another bowl, toss remaining chipotle with half of onions and the cheese; set aside 3/4 cup. Toss nopales with remaining cheese mixture in bowl. Lightly brush tortillas on both sides with remaining 1 tbsp. oil; sprinkle on one side with remaining 1/2 tsp. salt.
Set out 4 tortillas on a rimmed baking sheet. Sprinkle with half of nopales mixture. Repeat layers. Set remaining tortillas on top and sprinkle with reserved 3/4 cup cheese mixture.
Bake stacks until golden and crisp at edges, 8 to 10 minutes. Set on plates. Toss avocado mixture with lettuce and remaining onions, then mound on stacks.
BUY. Some supermarkets and Latino markets sell cactus with the spines, and others sell it de-stickered and ready to use--whole or cut up. When buying whole pads, look for ones that are smooth, crisp, and small to medium size.
PREP. To remove spines from pads, put on dish gloves. Scrape a knife almost flat down the length of both sides of pads to shave off spines and barbed, fuzzy dots. Pare rims with a vegetable peeler. Trim bases; rinse pads.
COOK. Once nopales hit the heat, they release a sticky juice (like okra), but it cooks off.
*Find corn-wheat tortillas such as La Tortilla Factory Hand Made Style at many regular grocery stores and at latortillafactory.com.