Step 1

Arrange a rack in the top third of the oven and heat the oven to 350°F.

Step 2

Trim and very thinly slice 1 pound Brussels sprouts with a knife, mandoline, or in a food processor fitted with the slicing blade (about 6 cups). Dice 1 medium shallot. Grate 3 ounces smoked Gouda cheese on the large holes of a box grater (about 3/4 cup). Finely grate 1/2 ounce Parmesan cheese on the small holes of a box grater (about 1/4 packed cup).

Step 3

Place 8 large eggs in a large bowl and whisk until blended. Add the Gouda, 1/3 cup whole or 2% milk, 1/2 teaspoon of the kosher salt, and 1/4 teaspoon of the black pepper. Whisk to combine.

Step 4

Heat 2 tablespoons olive oil in a 10-inch cast iron or oven-safe nonstick skillet over medium heat until shimmering. Add the shallot and sauté until softened, 2 to 3 minutes. Add the Brussels sprouts, 1/2 teaspoon smoked paprika, the remaining 1/2 teaspoon kosher salt, and the remaining 1/4 teaspoon black pepper, and sauté until tender and starting to brown in spots, 8 to 10 minutes.

Step 5

Spread the vegetables into an even layer, then pour the egg mixture over the top. Tilt the pan to make sure the eggs settle evenly over all the vegetables. Sprinkle with the Parmesan. Cook undisturbed until the eggs at the edges of the pan begin to set, about 2 minutes.

Step 6

Transfer the skillet to the oven and bake until the eggs are just set, 18 to 20 minutes. To check, cut a small slit in the center of the frittata. If raw eggs run into the cut, bake for another minute or two. If not, then the eggs are set.

Step 7

Heat the broiler to HIGH. Broil until the top of the frittata is lightly golden-brown in spots, about 2 minutes. Let cool in the pan for 5 minutes, then slice into wedges and serve warm or at room temperature.