In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and powdered sugar until pale and fluffy, about 2 minutes, stopping to scrape the sides of the bowl as needed. Beat in the egg, salt, cardamom, cinnamon, and vanilla.
In a medium bowl, whisk together the flours. With the mixer on low speed, gradually add the flour mixture, beating until just combined. Using lightly floured hands, divide the dough in half. Shape each half into a ½-inch thick disc and wrap it in plastic wrap. Refrigerate until firm, at least 2 hours.
Preheat oven to 350°F (180°C).
On a lightly floured surface, roll half of the dough out into a ¼-inch thick circle. (If the dough becomes too soft, rewrap it in plastic wrap and refrigerate until firm again.) Using a 2-inch cutter, cut the dough and place it about 1 inch apart on baking sheets lined with parchment paper. Using a 1-inch mini Christmas tree and/or star-shaped cutter, cut centers from half of the cookies. Reserve centers, and reroll with scraps once.
Bake until edges are lightly golden, 10-12 minutes. Allow cookies to cool on the baking tray for 5 minutes before removing to a wire rack to cool completely. Repeat with the remaining dough.
Spread about 1 teaspoon of jam onto the flat side of all solid cookies. Dust cookies with cutouts with powdered sugar, if desired. Place cookies with cutouts, flat side down, on top of jam and enjoy!