Migas With Green Salsa

Name: Migas With Green Salsa
Ingredients: 1 small yellow onion, very coarsely chopped; 3 garlic cloves; 1 serrano chile, halved; 8 oz. tomatillos (about 4 medium), husks removed, rinsed; 6 Tbsp. vegetable oil, plus more; 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, plus more; 1 small bunch cilantro, coarsely chopped; 2 Tbsp. fresh lime juice; 8 large eggs; 4 corn tortillas, halved, cut into ½-thick strips; 3 Tbsp. unsalted butter; 1 medium white onion, chopped; 3 oz. sharp cheddar, coarsely grated; Freshly ground pepper; Flour tortillas, sliced avocado, pickled sliced jalapeños, and lime wedges (for serving)