Preheat the oven to 350° F/175° C.

In the bowl of the stand mixer, mix together the butter and brown sugar until the mixture is light with no lumps.

Add the beaten egg, flour, salt, vanilla extract, and sesame seeds. Mix until all the ingredients are combined and the seeds are well distributed. If the mixture looks runny, sprinkle on a little more flour.

Drop the dough by the teaspoonful onto the prepared baking sheet. Dip the bottom of the glass into the ice water and press down on each ball of dough.

Bake in the preheated oven for 6 minutes or longer (mine took considerably longer). The cookies should be a dark golden brown and slip right off the pan. If they are stuck, pop the baking sheet back into the oven for a few more minutes.

Rest baked cookies on the sheet pan for a couple minutes, then transfer to a wire rack. Cool completely and store in a tin away from other cookies (to keep them crispy) for up to 2 weeks.