Preheat the oven to 375 °F. Line two baking sheets with parchment paper or a silicone baking mat.
In a small bowl, whisk together 5 ounces Rye Flour, 5 ounces all purpose flour, ½ teaspoon table salt and ½ teaspoon ground cardamom. Set the bowl aside.
In a mixer bowl, cream 8 ounces unsalted butter with 4 ounces granulated sugar until light and fluffy. Add the dry ingredients and mix just until combined. At this point the dough might look crumbly. Use your hands to knead the dough together. (See note)
Turn the dough out onto a lightly floured surface. Roll the dough to ⅛" thick. Use a 2½" round cookie cutter to cut as many cookies as you can. Re-roll the scraps and continue cutting until all the dough is used. If you plan to use the cookies as a tree decoration, cut a ½" hole off-center in each cookie. (See post content for more information.)
Line the cookies onto the prepared baking sheets, leaving ½" between cookies. (See note.) Lightly brush each cookie with egg white. Sprinkle lightly with caraway seeds and granulated sugar.
Bake until the cookies are just beginning to brown around the edges and the dough is set, about 8-10 minutes.
Slide the parchment paper or silicone mat onto a cooling rack and cool to room temperature.
The cookies will keep in an air tight container for about a week.