Preheat oven to 400°F. Toast nuts on a rimmed baking sheet until golden brown, about 4 minutes. Transfer to a small bowl. Toss shallots and vegetable oil on same baking sheet; season with salt and pepper. Roast until tender and browned in spots, 20–25 minutes. Let cool. Remove any leathery outer layers.

Meanwhile, bring vinegar and sugar to a simmer in a small skillet over medium-low, swirling occasionally, until syrupy, about 5 minutes. Stir in 1 Tbsp. butter; remove from heat. Arrange shallots, cut side up, in skillet, overlapping slightly if needed and filling in any gaps.

Roll out puff pastry on a lightly floured surface just to smooth any creases (if box has 2 sheets of pastry, stack and gently roll out to adhere). Cut out a circle that’s about 1" larger than the bottom of the skillet; prick in several places with a fork. Drape pastry over shallots, tucking edges inside skillet. Bake until pastry is golden brown and puffed, 25–30 minutes.

Meanwhile, heat remaining 2 Tbsp. butter in a medium skillet over medium-high. Cook mushrooms and garlic, tossing often, until mushrooms are tender and browned, 5–8 minutes. Season with salt and pepper; set aside.

Let tart sit until pastry is cooled slightly, 5–10 minutes. Carefully invert onto a plate. Top with burrata, mushrooms, arugula, Parmesan, and nuts. Drizzle with olive oil and lemon juice and season with salt.