Special equipment: a 9-inch springform pan
For the crust: Butter a 9-inch springform pan. Put the flour, sugar and 1/2 teaspoon salt in a food processor and pulse to combine. Add the butter and process until thoroughly combined with no visible chunks of butter. Add the egg and pulse until the dough just starts to clump. Transfer the dough to the springform pan and press it evenly on the bottom and 1 inch up the sides. Refrigerate for 30 minutes.
For the filling: Adjust an oven rack in the middle position and preheat the oven to 375 degrees F.
Toss the butternut squash slices with the melted butter, sage, 1 tablespoon honey and 1 teaspoon salt in a large bowl.
Sprinkle the Parmesan and breadcrumbs evenly on the bottom of the crust, then scatter over 1/2 cup of the gruyere cubes. Lay out a single layer of butternut squash slices, breaking off pieces of squash to fill any gaps. Scatter another 1/2 cup of gruyere over the squash. Top with the rest of the butternut squash slices and then the remaining 1/2 cup gruyere.
Put the springform pan on a baking sheet and bake, until the squash can be pierced easily with a knife, about 1 hour and 40 minutes to 1 hour 50 minutes. (If the crust gets too dark, cover loosely with foil.)
Meanwhile, combine the vinegar, crushed red pepper and the remaining 2 tablespoons honey in a microwave-safe bowl large enough to contain it (the honey may boil over in the microwave). Microwave for 30 seconds to heat up the honey and infuse the chile flavor. Reserve.
Drizzle the tart all over with the chile-infused honey and serve immediately.
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