Toss Brussels sprouts with olive oil in a rimmed baking sheet. Season with salt and pepper.
Roast in preheated oven for 25 minutes. Stir Brussels sprouts and continue baking until tender, about 15 minutes more.
Heat a skillet over medium-high heat; cook and stir prosciutto until crisp, about 5 minutes. Remove from heat.
Allow Brussels sprouts to rest for 5 minutes. Thread a Brussels sprout half on a toothpick, followed by a few chunks of prosciutto and another sprout half. Repeat with remaining sprouts and prosciutto.