Arrange brussels sprouts in a single layer on a baking sheet. Drizzle oil over the sprouts, and then evenly sprinkle salt and chili flakes.
Roast 16-18 minutes, until the edges darken and caramelize. Add cooked brussels sprouts to a large mixing bowl.
For hard, not fully ripe persimmons: Turn on the oven broiler. Arrange the persimmon slices on the baking sheet. Broil about 2 minutes, until warm. For soft, ripe persimmons: Skip broiling and add to the salad uncooked.
Add persimmons to the bowl with the sprouts. Add the pecans.
Whisk together all the maple vinaigrette ingredients. Toss the salad in the dressing, and serve immediately.
Make Ahead: For a cold salad, make up to 3 hours ahead of time. Wait to dress the salad until right before serving.