To Make Chimichurri Sauce: Combine parsley, cilantro, chile, vinegar, garlic, salt and pepper in food processor. Cover and pulse until chopped.
Add olive oil gradually, with food processor running. Continue to process until smooth. Makes 1 1/3 cup.
To Make Chimichurri Sandwiches: Combine chicken and 3/4 cup chimichurri sauce in large resealable plastic bag. Seal bag and turn to coat. Chill several hours or overnight.
Heat grill or grill pan over medium-high heat. Remove chicken from plastic bag, discarding any sauce remaining in bag. Grill chicken 2 to 4 minutes on each side, turning only once, until browned and no longer pink in center. Stack cooked chicken into 4 equal portions. Top each with 1 slice cheese. Remove from grill and let rest 5 minutes.
Heat canola oil in large skillet over medium heat. Add onion; saute 8 to 10 minutes or until golden brown and tender. Stir in salt and pepper.
Toast sandwich buns, if desired. Spread cut sides of buns with remaining Chimichurri Sauce. Serve chicken on buns topped with onions, avocado and lettuce. Serve immediately.