Beat the cream cheese, butter, granulated sugar, grated ginger and salt in a large bowl with a mixer until light and fluffy. Add the flour and mix until combined. Turn the dough out onto a sheet of plastic wrap and pat into a 1-inch-thick rectangle. Wrap in the plastic wrap and refrigerate until firm, about 1 hour.
Line 2 baking sheets with parchment paper. On a lightly floured surface, roll out the dough into a 1/8-inch-thick rectangle, then fold into thirds like a letter. Roll out again to about 1/3 inch thick. Cut into stars using 1 1/2-to-2-inch cookie cutters and place 1 inch apart on the prepared baking sheets. Reroll the scraps and repeat. (Refrigerate the dough if it gets too soft.)
Beat the egg and 1 tablespoon water in a small bowl. Lightly brush the cookies with the egg wash, then sprinkle with the crystallized ginger and coarse sugar. Refrigerate the cookies until firm, about 20 minutes.
Preheat the oven to 350 degrees F. Bake until the cookies are slightly puffed and golden brown, 15 to 20 minutes. Let cool 2 minutes on the baking sheets, then transfer to a rack to cool completely. Store in an airtight container up to 3 days.