Place racks in the center and upper third of your oven and preheat oven to 425 degrees F. Fill a saucepan large enough to comfortably hold the brussels sprouts with water and bring to a boil. Add 2 teaspoons of salt to the water. While the water comes to a boil, prepare an ice bath.
Add Brussels sprouts to the boiling salted water. Cook until the sprouts are tender, about 10 minutes. Drain the Brussels sprouts, then plunge immediately into the ice bath. Let cool 1 minute, then place the Brussels sprouts on a layer of paper towels and lightly pat dry.
Transfer the Brussels sprouts to the center of a large, rimmed baking sheet. Drizzle with the oil, and sprinkle with the remaining ½ teaspoon salt, black pepper, and red pepper flakes (if using). Toss to coat, then spread into an even layer. With the base of a drinking glass or coffee mug, gently but firmly press on each sprout to flatten it into a disk that is 1/2-inch or so thick.
Bake the Brussels sprouts for 15 minutes, then remove from the oven and carefully flip. Sprinkle the Parmesan over the top, then return the sprouts to the oven and continue baking until they are crisp on the outside and the cheese is golden, about 5 to 10 more minutes. If you would like the top more crisp, transfer the pan to the upper rack and turn the oven to broil. Let broil for 2 to 3 minutes, watching carefully so that the Brussels sprouts do not burn. Remove from the oven and squeeze the lemon juice over the top (if using). Enjoy hot.
TO MAKE VEGAN: Omit the Parmesan.
TO STORE: Place smashed sprouts in an airtight storage container in the refrigerator for up to 4 days.
TO REHEAT: Gently rewarm sprouts on a baking sheet in the oven at 350 degrees F until hot.
TO FREEZE: Store Brussels sprouts in an airtight freezer-safe storage container in the freezer for up to 3 months. Let thaw overnight in the refrigerator before reheating. Brussels sprouts can become soggy once frozen, but they'll recrisp decently in the oven.