Preheat the oven to 325 degrees F.
Cut the squash through the equator to create 3 pieces: a 1-inch thick ring and 2 end pieces. Seed the squash. Wrap each end piece in foil. Wrap the onions in another piece of foil. Bake the squash and onions for 1 hour, then unwrap and allow to cool.
Turn the oven up to 400 degrees F. Lightly grease 8 (4-ounce) ramekins or custard cups with butter or oil.
Peel the cooled squash and cut the flesh into cubes. Pulse the cubes in the bowl of a food processor until pureed, or mash vigorously with a fork. Transfer the squash to a large bowl, add the egg yolk, eggs, Parmigiano-Reggiano, mascarpone, nutmeg, salt and pepper and stir gently to combine. Beat the egg whites to soft peaks, then gently fold them into the squash mixture.
Divide the batter equally among the prepared cups, and place them in a baking pan large enough to hold them all without tipping. Fill the pan halfway with warm water, cover the whole pan tightly with aluminum foil, and bake for 35 to 40 minutes, or until a toothpick inserted in the center of the custard comes out clean. Keep warm.
While the squash bakes, place a 12 inch-saute pan over medium heat and add 3 tablespoons oil. When the oil smokes add the shiitake and the sage and cook until soft and slightly crisp at the edges, about 8 minutes WITHOUT SHAKING THE PAN. Remove and allow to cool.
Peel the reserved squash ring and cut the flesh into julienne strips. Place the squash in a small bowl, add 1 tablespoon of olive oil and salt and pepper, to taste, and toss well to coat.
To make the vinaigrette: pour the vinegar into a small bowl. Gently whisk in remaining 1/2 cup olive oil and season with salt and pepper.
To assemble the dish, carefully unmold 1 custard onto each of 8 warmed dinner plates, running a knife around the edge of the cup to loosen if necessary. Carefully halve the onions through the equator and place 1 half on each plate. Surround the custard with some of the mushroom mixture, garnish with the julienned squash and vinaigrette, and grate plenty of cheese over each plate before serving.
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