Preheat your oven to 400°F (200°C). Line a large, sturdy baking sheet with parchment paper or aluminum foil for easy cleanup. If using foil, lightly grease it with cooking spray or oil. Ensure the baking sheet is large enough so the sprouts can be spread in a single layer.

Wash the Brussels sprouts thoroughly. Remove any loose or damaged outer leaves. Trim the tough ends of the stems, then cut the sprouts in half lengthwise, from the top down through the root end. Try to keep the root end intact to prevent the leaves from separating during roasting.

In a large mixing bowl, toss the trimmed and halved Brussels sprouts with 2 tablespoons of the olive oil. Add the kosher salt and black pepper, and toss vigorously until every piece is lightly coated. This initial coating ensures the basic flavor is integrated.

In a separate, small bowl, create the Parmesan crust mixture: Combine the finely grated Parmesan cheese, the remaining 2 tablespoons of olive oil, garlic powder, onion powder, dried Italian seasoning, and the optional red pepper flakes. If using, stir the all-purpose flour into this cheese mixture now. The resulting consistency should be a thick, somewhat paste-like mixture.

Add the Parmesan crust mixture directly to the bowl containing the seasoned Brussels sprouts. Use your hands or a rubber spatula to carefully mix and coat each sprout half evenly. Ensure that the cut sides of the sprouts are heavily coated with the cheese mixture, as this is the side that forms the crust.

Transfer the coated Brussels sprouts to the prepared baking sheet. It is crucial to arrange them in a single, even layer, positioned so that the cut side (the side coated with the cheese) is facing down against the baking sheet. This direct contact is essential for developing the deep, golden, crispy crust.

Place the baking sheet in the preheated 400°F (200°C) oven and roast for 12 to 15 minutes without disturbing them. Do not open the oven door during this period; the cheese needs time to melt and crisp onto the pan. You should start to smell the cheese browning.

After the initial roasting interval, carefully remove the baking sheet. Using a thin, sturdy metal spatula, gently scrape under the sprouts and flip them over so the rounded side is now facing down. If the crust is sticking, gently pry it up, ensuring the crispy cheese remains attached to the sprout. Return the sprouts to the oven and roast for an additional 8 to 10 minutes, or until the sprouts are fork-tender and the edges of the green leaves are slightly charred.

Remove the baking sheet from the oven. Scrape any remaining crispy pieces of Parmesan crust off the sheet and mix them in with the sprouts. Serve immediately while they are hot and the crust is at its crispiest.