Heat a nonstick skillet over medium-high heat with EVOO, 2 turns of the pan. Add chorizo and shrimp and cook 3 to 4 minutes. Add sliced garlic and toss. Add piquillos and sherry or wine and let it absorb. Add juice of 1/2 lemon and turn off heat.
Heat a cast-iron skillet over medium to medium-high heat. Spray the tortillas or skillet lightly with oil. Blister a tortilla and flip. Top the tortilla with a layer of the shredded cheeses. Arrange chorizo and shrimp on half the quesadilla and fold over. Press with a spatula to set, turn a few times to melt the cheese and remove to a cutting board.
To a food processor bowl, add shallots, crushed garlic, 1/4 cup EVOO, vinegar, juice of 1/2 lemon, cilantro, parsley and season with salt and pepper. Pulse to process into sauce.
Cut quesadilla into wedges and arrange on platter with dollops of chimichurri, Greek yogurt or creme fraiche and pickled sliced jalapeno peppers.