Cut the crystallized ginger into large chunks, roughly 1/2 to 1 inch in size. Measure out 3/4 cup (about 95 grams) of the flour and set the remaining flour aside.

Working in two batches, process the crystallized ginger with the flour in a mini food processor. For each batch, add 60 grams of crystallized ginger chunks and half of the 3/4 cup flour (about 47 grams) to the food processor. Pulse until the ginger pieces are broken down to the size of small peas or lentils. You don't want them too fine—visible chunks add texture to the cookies.

The flour serves an important purpose here: it keeps the sticky crystallized ginger from clumping together and allows the processor to chop it more evenly. Transfer the ginger-flour mixture to a medium bowl and repeat with the second batch of ginger and flour.

Once both batches are processed, add the remaining flour to the bowl with the ginger mixture and stir to combine. Set aside.

Measure out all the ingredients before you start mixing them together. Place the butter, light brown sugar, granulated sugar, salt, ground ginger, and cardamom (if using) in the bowl of a stand mixer. With the paddle attachment, mix the ingredients on medium speed until the mixture is thoroughly combined and smooth, about 2 minutes. The mixture should be light and fluffy.

Stop the mixer and scrape the sides and bottom of the bowl and the paddle attachment. Add all the flour-ginger mixture at once. Turn the mixer to low speed and mix until all the flour is just combined. The dough will be crumbly at this point, with visible pieces of crystallized ginger throughout.

Increase the mixer speed to medium and mix until the dough has come together into large pieces, about 30 seconds to 1 minute. The dough should be soft and easy to handle, with a pleasant ginger aroma.

Divide the dough into two pieces, each weighing about 370 grams. Lightly flour a clean working surface. The flour will prevent the dough from sticking as you shape it.

Place one portion of the dough on the floured surface and gently press it into a rough rectangular shape, about 1 inch thick. Now, with your hands, start to gently roll the dough back and forth, applying even pressure. As you roll, the dough will begin to lengthen and form a log shape. Continue rolling and shaping the dough until it forms a smooth, uniform log about 1 1/2 to 2 inches in diameter. If the dough starts to crack, don't worry! Just gently press it back together. The crystallized ginger pieces may poke out slightly—this is perfectly normal and adds to the rustic appearance of the finished cookies.

Once you're happy with the shape of your log, wrap it in wax paper. Cut a sheet of wax paper that is long enough to wrap around the log with a little extra for overlapping. Place the log 2 inches from the long edge of the wax paper. Bring the long edge of the paper together over the top of the dough and smooth it so there are no gaps. Roll the log in the wax paper. You may need to occasionally stop and pull the wax paper tight if any gaps form. Twist the ends of the wax paper like a candy wrapper to secure the log.

Repeat this process with the second portion of dough. Now your dough logs are ready for chilling.

Chill the dough in the refrigerator for at least 1 hour and for up to 2 days. Chilling the dough will help the cookies hold their shape when baking and allows the ground ginger flavor to permeate throughout the dough.

Preheat the oven to 300°F. Line a half baking sheet with parchment paper.

Remove one log of cookie dough from the refrigerator. Lightly brush the log with water and then sprinkle generously with the turbinado sugar. Use a ruler to mark the log every 1/2-inch for cutting the cookie log. Slice the dough into rounds using a sharp knife. If the crystallized ginger makes cutting difficult, gently press any pieces back into place. Place up to 12 cookie rounds on a cookie sheet, spacing them about 1 inch apart.

Bake the cookies for 15-18 minutes. The bottom of the cookies will be lightly golden brown. The cookies will still feel slightly soft in the center, this is normal. Take the cookies out of the oven and allow them to cool for about 5 minutes on the cookie sheet before moving them to a cookie rack to completely cool. As they cool, the cookies will firm up.

At this point the cookies are ready to eat. The warm spices and sweet bursts of crystallized ginger make these cookies perfect with tea or coffee.

Store the cookies in an airtight container for up to 10 days. The flavors will actually deepen slightly after a day or two as the spices continue to meld.

Once your cookie dough logs are securely wrapped in wax paper, you can freeze them to bake later. First, chill them in the refrigerator for at least 1 hour so the log is firm. Place the wrapped logs in a large freezer bag or airtight container. This freezer bag will help prevent freezer burn and keep your dough fresh. Make sure to squeeze out as much air as possible from the freezer bag before sealing it tightly. Lay the bag flat in the freezer. This will help the logs maintain their shape and prevent them from getting squished by other items in your freezer.

Shortbread cookie dough can be stored in the freezer for up to 3 months. When you're ready to bake, take a log out of the freezer and let it thaw in the refrigerator for several hours or overnight until it's firm but pliable enough to slice. Slice and bake according to the recipe instructions. Add a minute or two to the baking time to account for the dough being chilled.