Separate the eggs and return the whites to the refrigerator. You will need them when you bake the cookies. Place the egg yolks and the sugar in the bowl of a stand mixer fitted with the whisk attachment. Beat, first on medium-low, then on high, for 4 to 5 minutes, until the mixture is very pale and fluffy. Drop in cubes of butter, two at a time, beating to incorporate them well before adding more. This will take several minutes, but don’t rush the process, or the mixture won’t whip properly. You are aiming for the texture of a soft, emulsified buttercream.
When all the butter has been incorporated, switch to the paddle attachment. Add the flour and salt all at once and mix on low speed to incorporate without kicking up any flour. Once the flour is mixed in, beat on medium high for 10 to 20 seconds to make sure everything is thoroughly combined. Scrape the dough onto a sheet of reusable wrap or plastic wrap, wrap tightly, and refrigerate overnight.
Remove the dough and the egg whites from the refrigerator and let them sit at room temperature for about 30 minutes to warm up slightly. Preheat the oven to 325° F (165° C). Line two rimmed baking sheets with parchment.
Lightly flour a clean work surface and lay the dough on it. Lightly flour the top of the dough and your rolling pin. Roll the dough out to a thickness of 3/8-inch (5-mm), in other words, just under 1/2 inch (12 mm). Use a 2 1/2 inch (6 cm) six-petal flower cookie cutter with a hole in the center to stamp out canestrelletti. If you don’t have a cookie cutter with a hole in the center, use a solid one and then use a zucchini corer or even the wide end of a stainless-steel piping tip to cut out the hole. Place the cookies on the prepared baking sheets, leaving about 1 inch (2 1/2 cm) between them. You should be able to fit 12 per sheet. Gather up the scraps and the punched-out centers and roll out more cookies, for a total of about 24.
Lightly beat the egg whites and brush the tops of the cookies. Bake the canestrelletti, one sheet at a time, in the middle of the oven, for 25 to 30 minutes, until they are set and golden, but not too brown. Transfer the baking sheets to wire racks to cool for 20 minutes, then transfer the cookies to the racks to cool completely. Serve them as is or with a dusting of confectioners’ sugar. Store canestrelletti in an airtight container for up to 2 weeks.