Pulse almonds and sugar in food processor until very finely ground. Add the butter and process until smooth. Scrape the dough off the inside of the bowl if needed. Add vanilla and almond extracts and pulse to combine. Add the flour and salt and pulse to make a soft dough.
Turn the dough out onto a large piece of wax paper and roll into a log about 15 inches long and 1 1/2 inches wide. Wrap in the wax paper and refrigerate at least 30 minutes.
Preheat oven to 325 degrees. Line 2 baking sheets with parchment paper.
Cut the chilled dough into 1/2 inch slices and roll by hand into balls. Space the cookies evenly on prepared baking sheets and bake until golden, rotating sheets once, 15 to 20 minutes.
Put the confectioner's sugar in a pie plate. Briefly cool the cookies on a rack, then toss them in confectioner's sugar until evenly coated. Return to rack, cool to room temperature and then toss again in the confectioner's sugar.