Combine cranberries, sugar, orange zest, orange juice, water, ginger, and salt for filling in a medium saucepan over medium-high heat, stirring to dissolve sugar. Lower heat and simmer, stirring frequently, until cranberries burst. Continue cooking, mashing fruit a bit with the back of a spoon, until thick as jam, about 30 minutes. Remove from heat to cool completely.

Sift flour, ginger, cinnamon, and salt for cookies together in a medium bowl. Set aside.

Mix butter and white sugar together in the bowl of an electric mixer fitted with a paddle attachment until just combined. Add the vanilla. Gradually add in flour mixture on low speed just until the dough starts to come together. Dump onto a piece of plastic wrap and flatten into a disk. Wrap and chill for 30 minutes.

Roll the dough ¼-inch thick and cut into 2-inch rounds with a plain or fluted cutter. With 1/2 of the rounds, cut a hole from the middle of each round into a desired shape. Place all cookies on a parchment-lined baking sheet and chill for 15 minutes.

While cookies chill, preheat the oven to 350 degrees F (175 degrees C).

Bake in the preheated oven until the edges just begin to brown or the cookies are crisp, 20 to 25 minutes. Remove from the oven; let cookies cool for 1 minute on the pan, then remove to a wire rack and allow to cool completely, 15 to 20 minutes.