In a large bowl, combine ground turkey, breadcrumbs, egg, garlic, parsley, mustard, salt, and pepper. Gently knead until just combined and form mixture into 4 patties.
Toss sliced onion with vegetable oil and season with salt and pepper to taste. Arrange onions in the center of a strip of tin foil and fold edges of foil to form a packet (or saute onions over medium-high heat on stovetop if cooking indoors)
Grease the grates of an outdoor grill well with oil and preheat to 375F. Alternatively if cooking indoors, grease a large skillet well and heat over medium-high heat. Place burgers in a single layer on preheated grill grates or in pan and cook burgers 8-10 minutes, flipping once until golden-brown and meat registers 165F.
Place foil packet with onions directly on grill grates and toss 4-6 minutes until onions are tender.
Assemble burgers by spreading whipped feta on top half of toasted buns. Place arugula on bottom half of toasted buns and top with a burger patty, tomato slice, and grilled onions. Serve immediately and enjoy!