Special equipment: 9-inch tart pan with removable base
Put an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Butter and flour the bottom and sides of a 9-inch tart pan.
Unroll the pie crust and arrange it in the prepared pan. Gently press the crust into the bottom and sides of the pan. Trim any excess crust from the top of the pan. With the tines of a fork, prick the pastry in several places. Using a pastry brush, coat the crust with the beaten egg white. Put the pan on a baking sheet and bake until the egg white has set, about 10 minutes. Allow the crust to cool for 10 minutes.
In a medium skillet, heat the oil over medium-high heat. Add the pancetta and cook, stirring frequently, until brown and crispy, about 6 to 8 minutes. Drain on a paper towel lined plate.
In a medium bowl, combine the beaten eggs, mascarpone cheese, Gruyere, green onions, salt, pepper, and pancetta. Mix until smooth. Pour the mixture into the cooled crust and bake until the mixture has set and the top is golden, about 18 minutes Cool the tart for 10 to 15 minutes then remove it from the pan to a serving platter.
Cut the tart into wedges and serve warm or at room temperature.
Cook's Note: The tart can also be made using a 10-inch tart pan. Roll out the dough to an 11-inch diameter circle before placing it in the pan. Bake the filled tart for about 23 minutes until the filling is set.