Halve and toast buns. In a second medium bowl, whisk together remaining vinegar, 1 TBSP olive oil (2 TBSP for 4 servings), salt, and pepper. Add arugula and some pickled shallot to bowl; toss to coat. Spread top buns with pesto mayo, then assemble burgers with patties, tomato slices, and as much remaining pickled shallot as you like. Serve with salad, potato wedges, and any remaining pesto mayo on the side.