In a large saucepan, boil the cider until reduced to 1/2 cup, about 15 minutes. Stir in the cream and boil until thickened, about 2 minutes. Pour into a bowl, then whisk in 2 tablespoons butter and the vanilla. Refrigerate for at least 1 hour.
Meanwhile, unfold the puff pastry on a floured surface. Roll out to a 10-by-12-inch rectangle. Refrigerate on a parchment-paper-lined baking sheet for 20 minutes.
In a large nonstick skillet, melt the remaining 2 tablespoons butter over medium heat. Add the apples, sprinkle with 1/4 cup sugar and cook, stirring, until golden, about 6 minutes. Remove from the heat and stir in the cinnamon. Spread on a baking sheet; refrigerate for 15 minutes.
Position a rack in the upper third of the oven and preheat to 375°. Spread the apples over the crust, leaving a 1-inch border. Using the parchment, lift the pastry edges and fold in to form a crust, pinching gently to adhere. Brush the crust with the egg wash; sprinkle with sugar.
Bake until golden, about 35 minutes. Transfer to a rack; let cool for 15 minutes. Cut into slices, top with ice cream and drizzle with the cider mixture.