Heat butter and olive oil in a deep skillet or cast iron pan over medium heat. Add garlic; cook and stir until golden brown, about 1 minute. Season chicken with salt and pepper. Add to the skillet and cook until browned on each side, no longer pink in the center, and juices run clear, 3 to 5 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Combine olive oil, onion, parsley, cilantro, lemon juice, garlic, and salt together in a food processor; pulse until finely chopped.
Spoon chimichurri sauce over chicken tenders; serve.