Shallot Tarte Tatin

Name: Shallot Tarte Tatin
Ingredients: 1 tablespoon pine nuts or chopped almonds; 6 large shallots, peeled, halved lengthwise; 2 teaspoons vegetable oil; Kosher salt, freshly ground pepper; ¼ cup balsamic vinegar; 1 teaspoon sugar; 3 tablespoons unsalted butter, divided; 1 package frozen puff pastry, thawed; All-purpose flour (for surface); 8 ounces mixed mushrooms (such as maitake, oyster, and/or shiitake), torn into bite-size pieces; 1 garlic clove, smashed; 3 ounces burrata, torn, or ricotta; 1 cup baby arugula; ½ ounce Parmesan, shaved; Olive oil and lemon wedge (for serving)