In a large bowl with a handheld mixer or a stand mixer fitted with the paddle attachment, combine the sugar with the lemon zest. Add the softened butter and cream well until light and fluffy.
Add the egg yolks one at a time and beat well after each addition. Add the rum or vanilla extract.
Sift the flour with the salt in a separate bowl and gradually add to the butter mixture. Mix until just combined but do not beat.
Shape the dough into a log about 62.5 inches (6 ⅓ cm) diameter, cover with plastic wrap and let sit in the fridge for at least one hour but preferably 3. You may also let it sit overnight and bake the next day.
When you are ready to shape the bussolai cookies, preheat the oven to 350°F (180°C). Slice the dough into equal pieces and, with the tips of your fingers, roll each into a rope about 8 inches (21 cm) long with a slightly tapered end. Shape into a circle 2 ¼ -¾ inch (which will be about 6-7 cm in diameter) and place on a baking sheet lined with parchment paper. Alternatively, shape the dough into S shapes.
Bake the bussolai cookies for about 14-17 minutes or until they start to turn golden at the edges.
Transfer to a wire wrack and let cool completely. Serve with sweet wine, tea or coffee.