Combine all burger ingredients, except the onion and lettuce, in a large bowl. Mix with your hands until just combined (don’t overmix) and divide into 4 large patties. Round them out and flatten slightly to about 1-inch thick, making a dent in the middle of each with your thumb.
Season the burger exterior with salt and pepper and refrigerate on a plate while you make the pesto. You can also turn on your grill at this point.
To make the pesto, combine all the ingredients except the olive oil and the feta in a food processor and blend until combined, and the pistachios form rice-sized pieces.
Slowly drizzle in the olive oil until you reach your desired consistency (keep it a little chunky) and then fold in some crumbled feta. Set aside.
Once the grill is hot, cook the patties on the indented side up and cook for 5 to 6 minutes per side until cooked to the desired doneness (5to 6 minutes per side is about medium).
Thinly slice the red onion and then prepare the buns. Assemble the burgers with chosen greens, then serve.