Dissolve the yeast in the water, add the sugar, and allow to sit until creamy, about 5 minutes. Beat in the egg and olive oil. Combine the flour and salt, and stir into the yeast mixture. Work the dough until it comes together in a coherent mass, adding flour as necessary. Turn out onto a lightly floured surface and knead for a few minutes, adding flour as necessary, until the dough is smooth; do not overwork. Shape into a ball. Place in a lightly oiled bowl, cover tightly with plastic wrap, and allow the dough to rise in a draft-free spot until doubled, about 1 hour.

Turn the dough out onto a lightly floured surface, gently knead a couple of times. For a galette, use all of the dough in one piece, for a double pie crust, cut into two pieces. Shape each piece into a ball and cover the dough loosely with plastic wrap. Let sit for 5 minutes. Then roll out into thin rounds, as directed in the recipe or use to line oiled pans. If not using right away, freeze the dough to prevent it from rising and becoming too bready. The dough can be transferred directly from the freezer to the oven.

Halve the squash and scrape away the seeds and membranes, then cut the flesh into 4-inch-square pieces. (If using butternut, cut it in half crosswise, just above the bulbous bottom part, then cut these halves into lengthwise quarters and scrape away the seeds and membranes.)

Steam the squash until tender, 15 to 20 minutes, then transfer to a colander and allow to cool and drain for another 15 minutes (butternut squash will not be watery). When the squash has cooled, peel and place in a bowl. Mash with a fork, a large wooden spoon, a potato masher or a pestle.

Heat the oil over medium heat in a large nonstick skillet and add the onion. Cook, stirring, until tender, about 5 minutes, then stir in the garlic. Cook until fragrant, another minute or two. Remove from the heat and toss with the squash, sage, rice, parsley, Gruyere and Parmesan cheeses, all but 2 tablespoons of the eggs, salt and pepper to taste. Taste and adjust seasonings.

Heat the oven to 190°C.

Oil a 10-inch or 12-inch tart pan with a removable bottom or springform pan.

Roll out two thirds of the pastry and line the pan with the edges overhanging. Fill the crust with the squash mixture. Top with the remaining pastry, then crimp the edges of the top and bottom pastry together. Cut 4 small slits in the top crust with a sharp knife, brush with the reserved beaten egg and place in the oven. Bake until golden brown, 45 to 50 minutes. Allow to rest for at least 10 minutes before serving. This can also be served at room temperature.