First of all, preheat grill or broiler. Take cactus paddles in bowl, and add a quarter cup of olive oil as well as half teaspoon of salt.
Keep grilling or broiling the paddles until the grill marks appear on each side. Make sure that the paddles have turned dark green with black patches. This will be done only a few minutes of time.
Set aside, and let it cool down to room temperature. Now, it is the time to cover, and chill it for an hour or two. A lot of people cover it overnight, which is not a bad thing at all.
Cut the cactus into small pieces, preferably half inch. Take a large bowl, and combine the cactus, tomatoes, onions, chiles, cilantro and cheese. The remaining half cup of oil and a teaspoon of salt along with pepper should also be added at this stage.
It is ready to serve on plates now. But the final touch is very important. The cactus must be lined up with the lettuce leaves. Garnish it with avocado slices sprinkled with cracked pepper garnish.