Combine the lime juice, onions and salt in a small nonreactive bowl, tossing occasionally. Let sit at room temperature until the onions are pink, about 15 minutes.
Heat the oil in a medium nonstick skillet over medium-high heat. Add the chorizo and cook, breaking up with a wooden spoon, until browned, about 3 minutes.
Preheat a waffle iron to medium-high. Brush one side of 2 tortillas with oil and lay dry-side up on a work surface. Scatter each with 1/3 cup cheese, then the pickled onions. Sandwich with the remaining tortillas and brush the tops with oil.
Place 1 quesadilla in the waffle iron, close gently (don't push down) and cook until golden brown and the cheese is melted, 4 to 6 minutes. Repeat with the remaining quesadilla. Cut the quesadillas into wedges and top with the chorizo. Serve with salsa, sour cream and avocado.
Copyright 2013 Television Food Network, G.P. All rights reserved