1. Preheat the oven to 400°F.

  2. To ready the pastry: Roll the pastry to an 11" x 17" rectangle. Trim the outside edges straight, and cut a 1" strip off each edge. Trim 1/2" off the end of each strip; reserve the strips. Place the rectangle on a parchment-lined baking sheet.

  3. Brush the outside inch of the rectangle with water and place the strips on top to make a border. Prick the inside of the pastry with a fork. Brush the border with beaten egg and bake the pastry shell for 10 minutes. Remove from the oven and reduce the temperature to 350°F.

  4. To make the filling: In a medium bowl, whisk together the crème fraîche (or sour cream), egg, Parmesan, thyme, salt, and pepper. Spread over the center of the tart inside the border.

  5. Heat the olive oil and butter in a large heavy skillet set over medium-high heat. When the butter melts and starts to foam, add the mushrooms and shallots and cook for 5 minutes, stirring once halfway through the cooking time. Remove from the heat and cool slightly.

  6. To make the garnish: Place the shallot rings in a small heatproof bowl. Combine the vinegar and sugar and heat to a simmer, on the stove or in the microwave. Stir until the sugar dissolves, then pour the mixture over the shallots. Stir briefly and set aside.

  7. Spoon the mushrooms over the filling and return the tart to the oven to bake for an additional 15 minutes.

  8. To serve: Sprinkle the top of the tart with the pickled shallots and shaved Parmigiano-Reggiano before slicing.

  9. Refrigerate any leftovers, well wrapped, for several days.