Place butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until smooth and very creamy, about 1 minute.
Add both sugars and the salt and beat on medium speed until smooth and velvety, about 1 minute.
Reduce the speed to low and add the egg yolks and extract, beating until incorporated, about 1 minute. Stop the mixer scrape down the sides of the bowl and the paddle with a rubber spatula.*Save the egg whites for making the royal icing Add the flour to the bowl. Drape a kitchen towel over the mixer and make 5 (2-second) pulses on low speed to start to incorporate the flour. Remove the towel and mix on low speed just until the flour disappears into the dough and the dough looks soft, clumpy, and moist, about 30 seconds. (If you still have some flour in the bottom of the bowl, stop the mixer and use a rubber spatula to work it into the dough.)
Scrape the dough onto a work surface, gather it into a ball, and divide it in half.
Place a ball of dough between 2 large pieces of parchment paper. Flatten the ball with our your hands into a disk, then use a rolling pin to roll the dough, turning it over frequently, until it's 1/4 inch thick. Transfer dough (paper and all) to a baking sheet; set aside. Repeat with the second ball and place on top of the first. Refrigerate until the dough is very firm, about 1 hour.
Heat oven to 350°F. Remove a sheet of dough from the refrigerator and place on work surface. Remove top sheet of parchment and place on a clean baking sheet. Using a cookie cutter, stamp out as many cookies as possible.
Transfer cookies to the parchment-covered baking sheet, spacing them 1 inch apart. Use an offset spatula to remove the cookies if needed. If the dough is getting too soft, return to the refrigerator to chill.
Save the scraps and re-roll between two new pieces of parchment paper and return to the refrigerator to chill until solid. The cut and repeat.
Bake until cookies are light brown on the bottom, lightly golden around the edges and pale on top, about 10-12 minutes. Let cookies rest 1 or 2 minutes before carefully transferring them to a wire rack with a wide metal spatula.
Repeat cutting, baking, and cooling with the remaining dough.
Once cool, decorate cookies with royal icing.