In a food processor, pulse the flour, powdered sugar, salt, ground ginger, candied ginger and candied orange peel, until the candied ginger and orange peel is finely chopped and evenly distributed.
With the processor running, drop the pieces of butter in, pulsing just until distributed.Add the vanilla and the milk. Process until the crumbs start to stick together.
Dump the dough out onto the counter, and knead a few times to bring it together into a smooth, supple ball of dough with your hands.
If you don't have a food processor, you can make this dough with a bowl and pastry cutter. You'll need to cut the butter in with the pastry cutter, then use your hands to work in the cream until you have a smooth dough. It's much easier and quicker with a food processor, though.
Shape the dough into a disk, wrap in plastic wrap and chill for 15 minutes.
Lightly flour a clean work surface, and dust the dough with flour as well. Roll out to about 1/4 inch thick or slightly thinner. Dip a cookie cutter in flour and cut as many cookies as you can. Gather up the scraps, re-roll the dough, and continue cutting cookies until you've used up all the dough. How many cookies you get will depend on the thickness of your dough and the size of your cookie cutters.
Line two baking sheets with parchment paper. Set the cut cookies on the baking sheets, 1 1/2 inches apart. Chill in the refrigerator for 1 hour. If you have room in your freezer, you can just freeze the pans for 30 minutes.
Preheat the oven to 350.Bake the chilled cookies on the center oven rack for about 11 1/2 minutes. They should still be pale in color, with flaky edges, no dark browning, and no appearance of raw wet dough in the middle. If your cookies are larger/thicker, they may need to bake a little longer.
Cool the cookies on the pan for 5 minutes, then transfer them to a cooling rack. Set the rack over the baking sheet, and brush the warm cookies with the glaze.