If using fresh cactus paddles that still contain tough bumps and thorns, wear heavy gloves and, using a sharp knife, shave off the bumps and thorns and discard. Remove the thick, fibrous base of the paddle and the tough outer edge of peel. Cut the paddles into 1/4-inch thin strips. It is not necessary to peel the entire paddle. (If using jarred or canned cactus, assemble the salad ingredients as described below after slicing the cactus into thin strips.)
Place the sliced cactus in a large saucepan and add enough water to cover by 4 inches. Add enough salt to make the water quite salty, and bring the water to a boil. Continue to cook until the cactus strips are tender, 10 to 20 minutes depending on the cactus. When tender, drain the cactus strips and rinse well under cool running water to remove any slime. Set aside to cool completely before proceeding.
In a mixing bowl combine the cactus paddles, tomato, onion, and cilantro and toss to mix. In a separate bowl whisk together the olive oil, vinegar, salt and oregano and pour over the cactus salad. Toss until thoroughly combined and allow to stand for a few minutes and up to 1 hour for the flavors to marry.
When ready to serve, line a serving plate or platter with the lettuce leaves and arrange the cactus mixture on top of the leaves. Sprinkle the queso fresco over the top, then garnish with the sliced radishes and pickled jalapenos. Serve immediately.
Yield: 4 to 6 servings