Place a rimmed baking sheet in the oven, then heat the oven to 425ºF. Meanwhile, peel the carrots and cut crosswise into 1 ½- to 2-inch pieces. Cut in half lengthwise if less than ¾ inch in diameter; otherwise cut in quarters lengthwise.

Place the carrots and 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 olive oil, kosher salt, and 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 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.