Preheat oven to 425°F. Bring a large pot of water to a boil over high heat. Add trimmed Brussels sprouts and cook until tender, about 10 minutes. Pour into a colander and let stand for 5 minutes.
Meanwhile, line 2 large baking sheets with parchment paper. Combine 1 cup Parmigiano-Reggiano and 1½ teaspoons garlic powder in a medium bowl. Sprinkle rounded teaspoonfuls of the cheese mixture into circles 1½ inches apart on the prepared baking sheets.
Place 1 Brussels sprout on each cheese mound. Smash the sprouts with the bottom of a sturdy cup or measuring cup to about ¼ inch thick. Sprinkle the sprouts with the remaining cheese mixture. Bake until the cheese on the bottom has melted and is golden brown and crispy, about 10 minutes. Carefully flip the sprouts, crispy-cheese side up, onto a serving platter. Sprinkle with ½ teaspoon salt.
