Pulse the almonds and sugar in a food processor until very finely ground. Add the butter and process until smooth, about 1 minute. Scrape the dough off the inside of the bowl, if needed. Add the 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 waxed paper and roll into a log about 15 inches long and 1 1/2 inches wide. Wrap and refrigerate for 30 minutes.
Preheat the oven to 325 degrees F. Line 2 baking sheets with parchment paper.
Cut the chilled dough into 1/2-inch pieces and roll by hand into balls. Space the cookies evenly on the prepared baking sheets and bake until slightly golden, rotating the sheets once, 15 to 20 minutes. Put the confectioners' sugar in a pie plate. Briefly cool the cookies on a rack, then gently toss in the confectioners' sugar until evenly coated. Return to rack, cool to room temperature, and then toss again in the confectioners' sugar.