Preheat oven to 450°F. Brush a 9-inch cake pan with 1 tablespoon oil; set aside.

Stir together sweet potatoes and salt in a large bowl. Let stand for 15 minutes; then place in a clean kitchen towel and squeeze as dry as possible.

Place back in large bowl and add egg whites; stir to combine. Press on the bottom and up sides of a 9-inch cake pan and gently brush with remaining ½ tablespoon oil.

Bake until starting to brown and crisp around edges, about 15 minutes. Remove from oven and set aside to cool slightly, 15 to 18 minutes. Reduce oven temperature to 325°F.

Meanwhile, place bacon in a large skillet and heat over medium. Cook, stirring occasionally, until rendered and crispy, about 12 minutes. Remove from pan to a paper towel-lined plate and set aside. Cool and discard bacon grease, or reserve for another use.

Whisk together eggs and cream in a large bowl until smooth. Whisk in 3 ounces (about ¾ cup) of the Gruyère, Parmigiano-Reggiano, whites and light greens of scallions, salt, pepper, and all but 1 tablespoon of the reserved bacon.

Pour filling into crust. Bake at 325°F until center is just set but slightly jiggly, about 35 to 40 minutes, topping with remaining 1 ounce (about ¼ cup) Gruyère after 15 minutes.

If desired, broil on high for the last 2 to 3 minutes until golden brown.

Let cool on a wire rack for 20 minutes. Garnish with dark greens of scallions and reserved bacon. Slice and serve warm or at room temperature.