Step 1

Preheat oven to 375°F. Line a large rimmed baking sheet with parchment paper. Gently roll out 1 sheet of frozen puff pastry, thawed (from a 17.3-ounce package) on a lightly floured surface to a 10" square (just enough to even out). Transfer to prepared sheet.

Step 2

Brush pastry with 1 large egg beaten with 1 teaspoon water. Arrange twelve 1/8"-thick rounds peeled butternut squash (cut from squash's neck) over pastry, overlapping as needed and leaving a 1/2" border. Place another sheet of parchment paper over squash. Set another large rimmed baking sheet over the tart. (This will weigh down the pastry dough and steam the squash slices.)

Step 3

Bake until bottom of pastry begins to brown and top begins to puff, about 10 minutes.

Step 4

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, 25-30 minutes longer.

Step 5

Meanwhile, combine 1/4 cup honey, 1 thinly sliced Fresno, jalapeño, or red Thai chile, and 2 tablespoons water in a small saucepan. Bring to a boil over medium heat (add another thinly sliced chile if more heat is desired). Boil until thickened slightly and syrupy, about 6 minutes.

Step 6

Line a plate with paper towels. Heat 2 tablespoons 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.

Step 7

Slice tart. Arrange 1/4 cup shaved Parmesan on top; drizzle with chile-infused honey. Garnish with fried sage leaves and a few grinds of black pepper.