In a mixing bowl, combine the butter, sugar, and salt. Cream together by hand or on medium speed with the paddle attachment. Cream until light and fluffy.
On medium speed, add the eggs one at a time while mixing. Cream until the eggs are fully incorporated. Scrape down the sides and bottom of the bowl if necessary.
Add the lemon zest, orange zest, and vanilla extract and stir until combined.
On low speed, or gently by hand, add the flour. Mix until the dough is well-combined.
Add the almonds and mix until just incorporated.
Divide the dough in half and chill for 15 minutes.
While the dough is chilling, mix the chopped almonds and sugar together.
Remove the dough from the fridge and roll each half into a ten-inch log. Use flour liberally while making this log, to stop the dough from sticking to your counter.
Brush each log with a little water and generously cover with the almond and sugar mixture.
Place on a parchment-lined baking sheet with 6 inches between each log.
Bake the mandelbrot for 35 to 40 minutes at 350° F. Remove from the oven and cool for 15 to 20 minutes.
Slice the mandelbrot logs into individual cookies about 1/2-inch thick.
Place the slices, standing up on their bottoms, back on a parchment-lined sheet pan and bake for 20 to 30 minutes. Remove from the oven and cool.
Notes
Store in an airtight container for up to 2 months.