These individual puff pastry tartlettes make a tasty and unusual starter course. For the prettiest effect, use just the neck of the butternut so you can create solid rounds of squash. The tartlettes are best eaten hot out of the oven, but they'll still taste great at room temperature.
Position a rack in the middle of the oven and preheat to 425 degrees F.
Peel the neck of the butternut, and then cut the squash crosswise into four 1/4-inch-thick rounds. Reserve the remaining butternut for another use.
Place the puff pastry sheets on a flour-dusted surface. Arrange the 4 butternut rounds on the pastry, leaving 1 inch between the squash rounds. With a sharp knife, cut around each disk of squash, leaving a 1/2-inch wide border of puff pastry. Transfer the pastry rounds to a parchment or Silpat-lined baking sheet. Use the remaining puff pastry to create inch-wide strips that can be used to build up the edges of each of the tartlettes. Moisten the outer inch-wide (exposed) edges of the pastry with a brush dipped in water, then paste the trimming strips onto the edges (single layer only). Prick the bottom of the pastry with the tines of a fork and refrigerate the pastry for at least 30 minutes.
Arrange one round of butternut on each pastry disk. Place 4 slices of goat cheese on top of each round of squash. Season with salt and pepper. Dip the sage leaves in the beaten egg (reserve remainder) and arrange them decoratively atop the cheese. Scatter some of the olives over each tartlette. Brush the top edges of the pastry with the beaten egg, taking care not to let the egg drip onto the baking sheet.
Bake until the edges of the pastry have puffed and turned golden, and the squash is tender when pierced with the tip of a knife, 12 to 16 minutes. Let cool for 5 minutes before transferring the tartlettes with a wide spatula to serving dishes.