Preheat the oven to 400 degrees F (200 degrees C). Line 2 sheets with parchment paper.

Combine confectioners' sugar, ground almonds, and baking powder in a large bowl. Add unbeaten egg whites one at a time, mixing until a firm dough forms. Knead dough for 2 minutes until it holds together and doesn't fall apart easily.

Roll dough into 18 ropes the thickness of your index finger, each rope a little longer than the next. Shape each rope into a ring. Arrange rings on the baking sheets.

Bake in the preheated oven until golden brown, about 10 minutes. Watch smaller circles carefully, as they may finish baking before the larger ones. Remove from the oven and let cool.

Mix confectioners' sugar and vanilla extract together to make icing. Pour icing into a piping bag fitted with a small tip.

Place a doily on a serving plate. Pipe a ring of icing as large as the largest cake ring. Place the largest cake ring on the icing. Pipe icing in a zig-zag pattern over the ring. Stack remaining cake rings on top, from largest to smallest, piping icing between each ring.