Cook the bacon in a 10-inch ovenproof skillet over medium heat until just crisp. Transfer bacon to a paper towel-lined plate wipe drippings from skillet. Crumble bacon.
Add 1 teaspoon of the olive oil and the shallots to skillet. Saute until soft and slightly golden, about 3 minutes.
Add the spinach and saute until wilted, about 1 minute. Transfer to a plate and let cool. Rinse and dry skillet.
Preheat the broiler. In a large bowl, whisk together the eggs, egg whites, milk, black pepper, salt, nutmeg, bacon, and 5 tablespoons of the Gruyere.
Return skillet to the stove. Add remaining olive oil. Pour in the egg mixture, stirring to distribute bacon. Arrange spinach and shallots over the top.