Preheat oven to 350 F.

Combine the flour and 1/4 teaspoon of salt in the bowl of a food processor fitted with a metal blade. Add the butter, and pulse, just until the mixture barely forms a coarse meal. Slowly drizzle in the ice water, and pulse once or twice, until the mixture comes together. Transfer to a liberally floured work surface and form a ball. Wrap in plastic wrap and press into a disk. Refrigerate for at least 30 minutes.

Peel the onions, and slice off only the root ends. Slice the onions in half, lengthwise, then in thirds, and set in a small baking dish. Drizzle with 2 tablespoons of the olive oil and the balsamic vinegar. Season with 1 teaspoon of the salt, and 1/4 teaspoon of the pepper. Roast for about 30-40 minutes, until the onions are tender but still hold their shape. Remove from oven and set aside to cool.

Over medium heat, melt the remaining 2 tablespoons butter in a large saute pan. Add the mushrooms, remaining 1/4 teaspoon of salt and 1/4 teaspoon of pepper. Saute 5 minutes, until they begin to turn golden. Remove from heat.

Unwrap the chilled dough and transfer to a lightly floured surface. Roll out into a 12-inch round. Transfer the dough to a pie plate, fold the overhang under, and crimp the edge. Pierce the bottom with a fork all over, and return to the refrigerator for another 20-30 minutes.

Combine the egg yolk, ricotta, last tablespoon of olive oil, 1/2 cup of the grated Parmesan, and the chives in a mixing bowl. Spread this mixture into the pie dish and settle the onions into the ricotta. Top with the mushrooms.

In a small bowl, beat the egg white with the heavy cream and last 1/4 cup of grated Parmesan. Pour this mixture over the filling in the pie plate. Cover with foil, and set on a rimmed baking sheet. Bake at 350 F. for 30 minutes. Remove foil and bake for an additional 15 minutes, until the egg has set and the crust is golden brown. Place on a wire cooling rack for at least 15 minutes before serving.

Serve warm or at room temperature.