Mix flour, baking powder, baking soda, salt, mozzarella, and rosemary a medium bowl. Working one at a time, dip prosciutto slices in the flour mixture before tearing them into small pieces and dropping them into the mixture, tossing well before moving on to the next slice (this will help prevent the sticky prosciutto from clumping together).
Whisk butter, egg, sugar, and buttermilk together in a separate medium bowl. Add dry ingredients to the wet, stirring until just combined.
Divide the batter evenly between 12 greased or paper-lined standard-size muffin tin cups.
Bake in an oven preheated to 375° F until pale golden, about 20 to 25 minutes. Cool briefly on a wire rack before removing muffins. Serve warm or at room temperature.