Make the salsa verde: Preheat the oven to 450 degrees F. Lay out the tomatillos, red onion and jalapeno on a baking sheet and drizzle with olive oil. Season with salt and pepper, then roast until slightly charred and softened, about 10 minutes. Remove from the oven and let cool slightly. Transfer to a food processor and puree until smooth. Add the cilantro and lime juice, season with salt and continue to puree until smooth. For a thicker salsa, pour into a cheesecloth-lined strainer and let drain.
Make the burgers: Preheat a grill or grill pan to medium high. Form the chorizo into 4 balls, then flatten with your palm. Season them with salt.
Oil the grill grates. Place the patties on the grill and press with a spatula to form thinner patties. Cook, without moving, until a good crust forms on the bottom, 3 to 4 minutes.
Flip and cook for another 2 to 3 minutes. Reduce the heat to medium low and top each burger with a slice of cheese; cover and let the cheese melt, about 1 minute. Remove the burgers and place on the toasted buns with the salsa verde, avocado, red onion and some piquillo peppers.