Preheat oven to 350°. On a piece of parchment paper big enough to line a rimmed baking sheet, draw a 10-in. circle with a pencil. Make 5 more and put all 6 aside. Set baking sheet in oven until hot. Combine cilantro and onion in a small bowl.
Set 1 piece of parchment paper on a work surface, pencil side down. Evenly scatter 1/2 cup plus 2 to 3 tbsp. cheese over circle (it won’t be covered completely). Chill remaining cheese. Remove baking sheet from oven and carefully set parchment paper with cheese on top.
Bake cheese until bubbling and starting to brown, 4 to 6 minutes. Scatter 2 tsp. cilantro-onion mixture on top. Rotate pan 180° and continue to bake until chicharrón is an even deep gold, 4 to 4 1/2 minutes more. Remove pan from oven.
Quickly blot fat from chicharrón with a paper towel. Slide a wide spatula under chicharrón to loosen all the way to center. Lift it with spatula and, using tongs, fold in half so edges meet; then gently fold in half again into a loose party-hat shape. If not yet crunchy, briefly hold in air with tongs, then stand it up on a cool baking sheet.
Repeat to make more chicharrónes, using a hot baking sheet and fresh piece of parchment for each one; to save time, stagger a second batch to go into the oven just after you sprinkle the first with cilantro-onion mixture.
Serve with salsas, guacamole, and tortillas if you like.
MAKE AHEAD Through step 5, cooled, draped with foil, and chilled up to 1 day; recrisp in a 300° oven until slightly warm, 45 to 60 seconds (no longer, or they’ll collapse).