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.
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