Melt butter in a large skillet over medium-high heat. Add pumpkin and apples, and cook, stirring occasionally, 8-10 minutes. Reduce heat to low, add sugar and vanilla beans, and scrape browned bits from bottom of skillet with a wooden spoon. Cover skillet, and cook until pumpkin and apples are very soft and jammy but still holding their shape, about 30 minutes. Set aside to cool. When cool, discard vanilla beans and stir in pecans.

Preheat oven to 350°. Roll out 1 sheet of puff pastry into a square on a lightly floured surface. Using an 11″ tart ring as a guide, cut out a circle and set aside. Roll out remaining sheet of pastry into a square; then ease pastry into the tart ring, and set on a parchment-lined baking sheet. Trim edges to overhang by 1⁄4″. Fill pastry with pumpkin mixture; then lay reserved pastry circle over filling. Tuck edge of top pastry between filling and bottom pastry, and brush surface with egg. Fold overhanging bottom pastry back over top pastry. Bake until deep golden, 35-40 minutes. Allow to cool slightly before serving.