Place the dry ingredients (flour, confectioner's sugar, rosemary, salt and parmessan) in the bowl of a food processor and pulse until combined.
Add butter and pulse until a soft dough forms. Dough should hold together when squeezed between your fingers. If needed, add water and pulse to combine.
Place half the dough on a sheet or plastic wrap, form it into a loose log along one side of the sheet. Roll the log, twisting the ends of the plastic wrap, until it is tight and compact, about 2-1/2" diameter. Do the same with the second half. (I find 2 shorter logs easier to work with, and easier to store in the frig.) Chill in the frig for at least one hour. As a make ahead, they can be stored in the frig until the next day.
Preheat the oven to 375F. Line 2 baking sheets with silicone baking sheets or parchment paper. Slice your dough logs into about 1/3" slices and place on baking sheets. Bake until the edges are just beginning to brown, @ 12-14 minutes. Do not overbake!
Cool for just a few minutes, then transfer to wire racks to cool completely. Shortbread can be stored in an airtight container at room temp, they can last for a couple of weeks or more.