Scrape down the bowl, add the egg and vanilla, and beat on medium-low until well combined, scraping down the bowl once or twice during the process, about 1 minute.
Add the cardamom, baking soda, salt, and flour and mix until the dough comes together.
Transfer the dough to a lightly floured surface and divide it into thirds (about 385g each). Keeping the pieces separate, return them to the bowl, cover lightly, and transfer to the refrigerator while you prepare the filling.
To make the jam filling: In a small bowl, mix together the jam and flour until no dry streaks remain. If you’re using different types of jam to make three flavors of pinwheel cookies, place 2 tablespoons (42g) of each jam in its own small bowl and add 1 tablespoon (7g) flour to each; stir to combine.
Transfer one piece of dough to a lightly floured piece of parchment (keep the other two pieces refrigerated) and shape into a rough rectangle. Using a floured rolling pin, roll the dough into a 1/4"-thick rectangle, about 12" x 7", with the long side facing you.
Spread 3 tablespoons (about 50g) of the jam mixture all over the dough, leaving a 1" border on the long side farthest from you. (If using three different flavors of jam, use the entire contents of one of the small bowls.)
Using the parchment for assistance, roll the dough away from you to make a log about 12" long and 1/2" wide. Pinch the seam shut, then use your hands to gently squeeze and roll the log to make it an even thickness.
Fold the parchment away from you to encase the log. Use your non-dominant hand to press a straight object (a ruler or a bench knife works well) in between the dough and the work surface, then use your dominant hand to gently tug the top piece of parchment away from you. This should pull the dough so that it butts up against the straight object and makes a very smooth, symmetrical log. See this blog post, How to make round cookies, for a visual demonstration.
Once you have an even log, spread the sparkling sugar on the parchment in a 2"-wide strip about 10" long. Roll the dough log in the sugar to coat evenly.
Wrap the log in parchment or plastic wrap, or transfer it to a Slice-and-Bake Cookie Dough Keeper. Refrigerate for at least 3 hours or up to 2 days.
Repeat the rolling, filling, and shaping process (steps 6 through 11) with the other two pieces of dough to make a total of three logs.
When ready to bake, preheat the oven to 375°F with racks in the upper and lower thirds.
Remove one log of dough from the refrigerator, unwrap or remove it from the Keeper, and slice it into 1/4"-thick rounds. Rotate the log about 90° (a quarter turn) every three to four slices to maintain a neat round shape. Transfer the slices to the parchment-lined baking sheets, about 15 per sheet (cookies can be close together; leave about 1/2" between each).
Bake the pinwheel cookies, two sheets at a time, for 12 to 14 minutes, rotating the sheets from top to bottom and front to back halfway through, until the edges are just beginning to turn golden. (If baking one sheet at a time, check cookies at 10 minutes.)
Slice and bake the remaining cookies.
Storage information: Store pinwheel cookies in an airtight container for up to 5 days. Freeze for longer storage.
