In a food processor, combine parsley, cilantro, garlic, red wine vinegar, oregano, red pepper flakes, and salt. Pulse until finely chopped, about 8 to 10 pulses. With the machine running, drizzle in the olive oil through the feed tube until the sauce is emulsified but still slightly chunky. Transfer to a bowl and set aside. The sauce will taste sharp now but mellows as it sits.

Pat the chicken completely dry with paper towels. This is crucial for good browning. Brush both sides with the 1 tablespoon olive oil, then season evenly with kosher salt and black pepper.

Heat your grill or grill pan to medium-high, about 400 to 450 degrees F. The grates should be hot enough that you can only hold your hand 5 inches above for 2 to 3 seconds.

Place the chicken on the grill and cook without moving for 5 to 6 minutes until release marks appear and the chicken releases easily. Flip and continue cooking until the internal temperature reaches 165 degrees F at the thickest part, about 5 to 7 minutes more for thighs, or 6 to 8 minutes for breasts. Transfer to a plate and let rest for 5 minutes to allow juices to redistribute.

Slice the chicken against the grain into 1/2-inch strips. Arrange on a platter and spoon about half the chimichurri over the top. Serve the remaining sauce on the side for guests to add as they like.