Add the almonds to a food processor and pulse until very finely ground, like coarse sand (not a paste). You should have about ½ cup of ground almonds.
In a medium bowl, beat butter and confectioners’ sugar with an electric mixer until smooth and creamy, about 2 minutes. Mix in vanilla, almond extract, and salt.
Gradually add the flour, mixing just until combined.
Gently fold in the ground almonds; do not overmix. If the dough feels dry or crumbly, knead lightly with your hands until it comes together.
Cover the bowl and chill the dough in the refrigerator for 1 hour. Chilling makes the dough easier to shape.
Preheat oven to 325°F. Line a baking sheet with parchment paper.
Scoop about 1 tablespoon of dough per cookie and shape each into a 2-inch crescent. (I made 17 cookies.) Place them 2 inches apart on the prepared baking sheet.
Bake 14-16 minutes, until bottoms are lightly golden but tops remain pale.
Let cookies cool on the baking sheet for 5 minutes, then roll gently in confectioners’ sugar while still slightly warm.
Once fully cooled, dust the cookies with a second layer of confectioners’ sugar before serving.