How to Make It
In a large skillet, melt the butter over high heat. Add the diced pears and cook, stirring occasionally, until they are golden around the edges, about 5 minutes. Add 1/2 cup of the sugar, the lemon juice and the vanilla bean and seeds and cook until the pears are tender and the liquid in the skillet has reduced to about 2 tablespoons, 4 to 5 minutes. Spread the diced pears on a baking sheet and let them cool, about 25 minutes. Discard the vanilla bean.
Lightly flour a work surface. Roll out the puff pastry to a scant 1/8-inch thickness. Using a plate as a template, carefully cut out four 7-inch rounds. Transfer the rounds to a large baking sheet lined with parchment paper (they will overlap slightly).
Preheat the oven to 425°. Spoon one-fourth of the pear filling over half of a pastry round, leaving a 1/2-inch border at the edge. Brush the border lightly with the egg-white wash. Fold the top half of the round over and press the edges together firmly to seal. Repeat with the remaining pear filling and pastry rounds. Refrigerate the hand pies for 30 minutes. Reserve the remaining egg-white wash.
Brush the tops of the hand pies lightly with the remaining egg-white wash. Sprinkle the tops all over with the remaining 2 tablespoons of sugar. Bake the hand pies for 15 minutes in the middle of the oven, then reduce the oven temperature to 400° and bake the hand pies for about 15 minutes longer, until they are a deep golden brown color. Let the hand pies cool for 20 minutes, then serve them warm with the vanilla ice cream.