Set your oven to 425°F and let it get fully hot. A properly heated oven gives you browning instead of steaming. Line your baking sheet with parchment paper while you wait.
Rinse the sprouts and pat them very dry. Slice off any dry or woody stem ends and pull away loose or yellow outer leaves. Wet sprouts steam in the oven, so dry them well.
Slice each sprout in half from top to bottom. Leave very small ones whole so they cook at the same pace as the rest. Even sizes mean even roasting.
Add the sprouts to a large bowl with the olive oil, salt, and pepper. Toss until every piece looks lightly glossy. Don't drown them, because too much oil makes them greasy.
Spread the sprouts on the lined baking sheet with the cut sides facing down. That flat side is what gets deeply golden. Leave a little space between each one.
Slide the pan into the oven and roast for 20 to 30 minutes. Start checking around 20 minutes, since sprout size and oven heat vary. They are ready when tender inside and browned around the edges.
While the sprouts are still warm, toss them with the lemon juice, zest, Parmesan, and thyme. Finish with parsley and a pinch of red pepper flakes. The residual heat softens the cheese and brings out the aroma.
Move the sprouts to a warm bowl or platter and taste for salt. Serve them hot for the best crunch. They lose crispness as they sit, so don't wait too long.