Place shortening 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 (if using) beating until incorporated, about 1 minute. Stop the mixer scrape down the sides of the bowl and the paddle with a rubber spatula.
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 or waxed 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 1/8 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 round 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. Save scraps to reroll later.
Bake until cookies are light brown on the bottom, lightly golden around the edges and pale on top, about 12 minutes. Let cookies rest 1 or 2 minutes before carefully transferring them to a cooling rack with a wide metal spatula.
Gather scraps, press into a disk, and roll between sheets of parchment or waxed paper. Place on a baking sheet and refrigerate until firm. Repeat cutting, baking, and cooling with the remaining dough.