Mix together brown sugar, cinnamon, nutmeg, ginger, cornstarch, and salt in a small bowl.
In a large bowl, toss peaches in lemon juice and vanilla extract. Coat peaches in brown sugar mixture and allow peaches to sit for 10 minutes.
Place a large piece of parchment paper onto a work surface and sprinkle with flour. Place thawed puff pastry sheet onto parchment paper and gently roll out to smooth the edges.
Place peach slices onto puff pastry, leaving a ~2-inch border on the sides. Wrap/pleat edges of puff pastry around the peaches.
Beat an egg in a small bowl and brush the edges of the puff pastry. Sprinkle with coarse sugar (optional).
Bake in preheated oven for 25-30 minutes until sides of puff pastry are browned and peaches are softened.
Slice like a pie and serve warm with vanilla ice cream.