Preheat oven to 375 degrees F.
Place the puff pastry on a piece of parchment paper and gently roll it out to a 10-inch square. Transfer it to a large rimmed baking sheet.
Create an egg wash by whisking together the egg and 1 teaspoon of water. Brush the pastry with the mixture.
Arrange the butternut squash pieces over the pastry, overlapping as needed and leaving a ½-inch border. Now we are going to parbake the squash. Place another sheet of parchment paper over squash then set another large rimmed baking sheet over the tart. This will weigh down the pastry dough and steam the squash.
Bake until bottom of pastry begins to brown and top begins to puff, about 10-15 minutes
Remove top baking sheet and discard top sheet of parchment paper. Brush squash slices with 1 tablespoon olive oil and season with kosher salt. Return tart, uncovered, to oven and bake until pastry is deep golden brown and cooked through, 20-25 minutes longer.
Meanwhile, combine ¼ cup honey, 1 thinly sliced jalapeno, and 2 tablespoons water in a small saucepan. Bring to boil over medium heat. Boil until thickened slightly and syrupy, about 6 minutes.
Line a plate with paper towels. Heat 2 tablespoons of olive oil in a small skillet until just beginning to smoke. Add 12 fresh sage leaves; fry until crisp, about 30 seconds. Transfer to paper towels to drain.
Cut tart into 9 pieces. Top with Parmesan and drizzle with spicy honey. Garnish with fried sage leaves and a few grinds of black pepper.