Preheat oven to 450 degrees. Wash and dry all produce. Peel and thinly slice shallot. Zest and quarter lemon. In a small bowl, combine half the shallot, juice from 2 lemon wedges, 2 TBSP hot water (1/4 cup for 4 servings), and 1/4 tsp sugar (1/2 tsp for 4); set aside. Cut potatoes into 1/2-inch-thick wedges. Toss on a baking sheet with a large drizzle of olive oil, salt, and pepper. Roast on top rack for 15 minutes.

Once potatoes have roasted 15 minutes, remove from oven and push to one side of sheet. Mound Parmesan on empty side in two even piles. (For 4 servings, leave potatoes roasting and add cheese to a second sheet; bake on middle rack.) Return to oven and bake until cheese is melted and crispy at edges and potatoes are tender, 5-7 minutes more. Let frico cool for 1 minute on sheet, then transfer to a plate.

Meanwhile, trim and thinly slice mushrooms. Heat a large drizzle of oil in a large, preferably nonstick, pan over medium-high heat. Add mushrooms; cook until golden brown, 3-4 minutes. Add remaining shallot; cook, stirring, until softened, 2-4 minutes. Stir in 1/4 cup water (1/3 cup for 4 servings) and stock concentrate; cook until liquid has mostly evaporated, 1-2 minutes. Season with salt and pepper. Turn off heat; transfer to a plate. Wipe out pan.

Meanwhile, halve buns. In a medium bowl, combine beef and 1/4 tsp truffle zest (1/2 tsp for 4 servings); season generously with salt and pepper. Form into two equal-sized patties (four patties for 4), each a bit wider than a burger bun.

Heat a drizzle of olive oil in pan used to cook mushrooms over medium-high heat. Add patties and cook to desired doneness, 3-5 minutes per side. Meanwhile, in a second small bowl, combine mayonnaise, sour cream, 1/4 tsp truffle zest (1/2 tsp for 4 servings), salt, and pepper. (Taste and add more truffle zest if desired.) Toast buns until golden.

Drain pickled shallot; toss in a second medium bowl with mixed greens, juice from remaining lemon, a drizzle of olive oil, salt, pepper, and lemon zest to taste. Spread cut sides of buns with a bit of truffle cream. Fill buns with patties, mushroom mixture, and frico. Serve with potato wedges, salad, and remaining truffle cream on the side.