Preheat your oven to 325°F and line two baking sheets with parchment paper.
In a large bowl, beat the softened butter with the sifted powdered sugar and salt until light and fluffy.
Add the finely ground almonds and vanilla extract, stirring until just combined.
Gradually incorporate the all-purpose flour until a soft dough develops. If the dough is too dry, add a splash of water to help it come together. If it feels too soft, refrigerate it for about 10 minutes to make it easier to handle.
Using a small spoon or cookie scoop, form 1-inch balls and place them about 1 inch apart on the prepared baking sheets.
Bake for 15-20 minutes until the edges are lightly golden. Remove the cookies, and while they are still slightly warm, roll them in extra powdered sugar. If desired, cover them with two layers of powdered sugar for full coverage. Transfer to a wire rack to cool completely before serving.