6 tablespoons unsalted butter
4 pounds Pink Lady and/or Granny Smith apples, peeled, cored, cut into 1-inch pieces
2/3 cup (packed) dark brown sugar
3 tablespoons all-purpose flour
1 teaspoon finely grated lemon zest
2 tablespoons fresh lemon juice
1 tablespoon vanilla extract
2 teaspoons ground cinnamon
1 (1/2-inch) piece ginger, peeled, finely grated
1/2 teaspoon kosher salt
1 package frozen puff pastry (preferably Dufour), thawed, cut into irregular 1-inch pieces
Granulated sugar (for sprinkling)
2 pints vanilla ice cream

Preheat oven to 425F. Cook butter in a small saucepan over medium, swirling often, until bubbling and golden, about 5 minutes; set aside.
Toss apples, brown sugar, flour, lemon zest, lemon juice, vanilla extract, cinnamon, ginger, and salt in a large bowl.
Transfer to a shallow 3-quart baking dish. Drizzle all but 2 Tbsp. brown butter over apples.
Top with pieces of puff pastry, then brush pastry with remaining brown butter. Sprinkle with granulated sugar.
Bake until pastry is puffed and golden around edges, 2530 minutes.
Reduce temperature to 350F and bake until juices are thick and bubbling and pastry is brown all over, 3035 minutes longer.
Using a spoon, press pastry down into warm juices (but dont submerge); let cool slightly.
Serve warm with ice cream.
Do Ahead
Pandowdy can be made 1 day ahead. Let cool. Cover loosely with foil and store at room temperature.
Reheat uncovered in a 200F oven 20 minutes before serving.