Using a rubber spatula, mix together the butter and 1 1/2 cups of the Parmesan in a medium bowl. Spread the mixture evenly on the cut sides of the baguette quarters, then sprinkle with the remaining 1 cup Parmesan.
Place the baguettes, cut-side up, on a rimmed baking sheet. Broil until golden brown and bubbly, about 3 minutes. Remove to a cutting board and allow to cool.
Pile the prosciutto cotto on top of the baguettes. Slice each quarter into 4 pieces. Serve open-faced.