Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Cut potatoes into 1/2-inch-thick wedges. Halve, peel, and thinly slice shallot. Slice tomato into 1/2-inch-thick rounds.

In a small microwave-safe bowl, combine half the vinegar, 1/2 tsp sugar (1 tsp for 4 servings), salt, and pepper. Stir in shallot, then cover with plastic wrap. Microwave for 1 minute. Remove plastic wrap and set aside, stirring occasionally, until ready to serve.

Toss potatoes on a baking sheet with a large drizzle of olive oil, salt, and pepper. Roast on top rack until golden brown and tender, 20-25 minutes. Meanwhile, in a small bowl, combine mayonnaise, sour cream, and half the pesto. Season with salt and pepper.

In a medium bowl, combine beef and remaining pesto; season generously with salt and pepper. Form mixture into two 1/2-inch-thick patties (four patties for 4 servings). Divide mozzarella between centers of each patty. Fold edges of meat around cheese, shaping and sealing to create cheese-stuffed patties.

Season tomato slices all over with salt and pepper. Heat a drizzle of olive oil in a large, preferably nonstick, pan over medium-high heat. Add tomato slices and cook until lightly browned and slightly softened, 1-2 minutes per side. Transfer to a plate. Add another drizzle of olive oil to same pan; add patties and cook to desired doneness, 3-5 minutes per side.

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.