Place a rimmed baking sheet in the oven, then heat the oven to 425ºF. Meanwhile, peel 12 ounces medium carrots and cut crosswise into 1 1/2- to 2-inch pieces. Cut in half lengthwise if less than 3/4 inch in diameter; otherwise cut in quarters lengthwise.

Place the carrots and 2 tablespoons water in a large microwave-safe bowl. Cover tightly with plastic wrap, leaving a portion open to vent. Microwave on HIGH until the carrots are knife-tender, 5 to 6 minutes. Carefully uncover the bowl to release the steam.

Drain off any residual liquid. Add 1 tablespoon olive oil, 1/4 teaspoon kosher salt, and 1/4 teaspoon black pepper, and toss to coat.

Arrange the carrots in a single layer on the hot baking sheet. Use the bottom of a glass or metal measuring cup to smash the carrots until they crack and flatten. Sprinkle evenly with 3 tablespoons grated Parmesan cheese.

Roast until the cheese is melted and browned, and the carrots are crispy, 13 to 15 minutes. Let cool for 5 minutes on the baking sheet to continue to crisp before serving.