Step 1Make the salsa: In a large bowl, combine the tomatillos, cucumber, radishes, cilantro, red onion, jalapenos, lime zest and juice, olive oil, cayenne pepper and salt. Toss to combine, then transfer to a nonreactive container, cover and refrigerate overnight.
Step 2Warm the tortillas. Heat the oven to 350 degrees. Wrap the tortillas in a damp towel and place the towel on a baking sheet. Heat the tortillas just until warmed through.
Step 3Heat an 8- to 9-inch diameter nonstick skillet over medium-low heat. Scatter one-half cup grated cheese in a circle, about 7 inches in diameter. As the cheese melts, spread it evenly across the pan so it cooks evenly, spreading the cheese further out to the edges of the pan as it melts. Rotate the pan as needed so the cheese cooks evenly. Once the cheese starts to crisp on the bottom and lightly color on the top, 6 to 7 minutes, blot dry and continue cooking until golden brown, another minute or two.
Step 4Using a plastic spatula, remove the cheese from the pan. Immediately form the cheese into a cone or drape it over a rolling pin, using a spatula and being careful, as the cheese is quite hot. It will cool very quickly, becoming crunchy and keeping whatever shape you form it into. Cool on paper towels and repeat until all of the cheese has been used. Serve the chicharrones with the warm tortillas and a bowl of salsa.