Spanish-Style Chorizo and Shrimp Quesadilla with Chimichurri

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2 tablespoons EVOO, plus about 1/4 cup for chimichurri

8 ounces Spanish-style chorizo, casing removed and sliced, preferred brand Gaspar's Hot 

8 ounces medium shrimp, halved along the devein line 

5 cloves garlic, 4 thinly sliced or grated, 1 crushed  

4 piquillo peppers (in water, drained) thinly sliced, or one 4-ounce jar chopped pimiento, drained

About 1/3 cup dry sherry or white wine 

1 small lemon, halved 

2 large flour tortillas 

Cooking spray, non-aerosol, olive oil 

1 cup shredded Pepper Jack 

1 cup shredded Manchego 

1 shallot, peeled and coarsely chopped 

2 tablespoons sherry vinegar or wine vinegar

1 cup each parsley and cilantro, loosely packed 

Salt and pepper

Greek yogurt or creme fraiche

Pickled sliced jalapeno peppers