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 the grated Parmesan cheese.