The best part of this twist on pigs-in-a-blanket isn’t the roasted Brussels sprouts, buttery puff pastry or even the pleasantly salty prosciutto: it’s the Parmesan cheese that is cleverly pressed into the dough while rolling and then sprinkled on top before baking. There’s umami goodness throughout and toasty cheesiness all around.
Preheat the oven to 425ºF. Spread the Brussels sprouts out on a rimmed baking sheet, drizzle with the oil and sprinkle generously with salt and pepper. Bake, stirring the sprouts once halfway through cooking, until golden brown and tender, 18 to 20 minutes. Transfer the baking sheet to a rack and let cool completely. Leave the oven on.
Sprinkle the Parmesan evenly on a work surface and lay the pastry sheet over it. Roll the dough into a 12-inch square, sweeping up any loose cheese onto the dough and pressing it in place. Use an offset spatula or table knife to spread the butter over the pastry so it covers it from edge to edge in a thin, even coating. Lay the prosciutto over the butter so it covers the pastry completely. Cut the pastry square in half to make two 12-by-6-inch rectangles. Cut each rectangle crosswise into eight 1 1/2-inch-wide, 6-inch-long strips. Then cut each strip diagonally lengthwise so you have 32 long, narrow triangles. Place a Brussels sprout at the wide end of each triangle, then roll it up in each strip toward the tip (it will look like a crescent roll with the Brussels sprout peeking out on each end).
Transfer each bundle to a parchment-lined baking sheet and space 1 inch apart. Rake up all the excess Parmesan on the work surface and sprinkle it on top of the rolls. Bake until the pastry is golden brown and the Brussels sprouts are warmed through, 16 to 18 minutes. Transfer to a serving dish.