Beat the softened butter, sugar and salt in a 4-quart / 3.8 L bowl with a hand or stand mixer on medium speed for about 3 minutes, until pale and creamy.

Beat in the egg yolks one at a time, then the rum, lemon zest and vanilla, until the mixture looks smooth.

Add the flour and mix on low speed just until a firm, slightly sandy dough forms that isn't sticky. Press it into a disk, wrap it and chill for 30 minutes. Heat the oven to 170 C / 340 F during the last 15 minutes of chilling and line two 33 x 46 cm / 13 x 18 inch sheet pans with parchment.

Pinch off 20 g pieces of dough, about 30 in all. Roll each on an unfloured counter into a 15 cm / 6-inch rope about 1.2 cm / 1/2 inch thick, overlap the ends by 1 cm and press to seal into a ring about 4.5 cm / 1 3/4 inches across. If the dough feels stiff from the fridge, let it sit for 5 minutes first. Place 15 rings on each pan, 2.5 cm / 1 inch apart.

Bake both pans together on the upper and lower oven racks for 16 to 18 minutes, swapping and rotating the pans at 8 minutes, until the bottoms and edges are light gold and the tops are still pale.

Cool the rings on the pans for 5 minutes, then move them to a wire rack for 10 more minutes. They crisp as they cool. Store in an airtight tin once completely cold.