2tablespoonsbalsamic vinegarpreferably white balsamic, divided
Kosher salt and freshly ground pepperto taste
2large yellow or Vidalia onions
8loin lamb chopsabout 1-inch thick
1cupflat-leaf parsley leaves
Combine 2 tablespoons of the olive oil, the white wine, 1 tablespoon balsamic vinegar and salt and pepper in a shallow baking dish. Peel the onions, and cut them crosswise into 1/2-inch slices. Place them in the baking dish, along with the lemon halves, turn to coat with the marinade (it’s fine to stack the onion slices), and set aside.
Season the lamb with salt and pepper. Grill the onions and the lamb chops for about 8 minutes, about 4 minutes on each side, so that they get nicely browned on both sides, turning the onions with a grilling spatula (you can use a spatula or tongs for the lamb). At the same time, grill the lemons cut side down for about 5 minutes. The internal temperature of the lamb chops should be 130°F for medium rare.
Remove the chops and the lemons from the grill and let sit on a cutting board for 10 minutes. Meanwhile, move the onions to a cooler area of the grill or turn the gas down, and let them continue to soften while the lamb and lemons sit, watching carefully to make sure the onions don’t burn.
While the chops sit and the onions finish cooking, place the arugula and parsley in a medium bowl. Drizzle with the remaining 2 tablespoons olive oil, 1 tablespoon vinegar, then season with salt and pepper and toss. Transfer the greens to a serving platter.
Place the lamb chops and onions on the greens. Nestle the lemon halves into the chops and onions, for people to squeeze over.