Preheat the oven to 400 degrees F (200 degrees C). Line 2 baking sheets with parchment paper.
Combine confectioners' sugar, ground almonds, and baking powder in a large bowl. Mix in egg whites, one at a time, to form a firm dough. 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 prepared baking sheets.
Bake in the preheated oven until golden brown, about 10 minutes. Watch smaller cookie rings carefully, as they may finish baking before the larger ones. Let cookies cool completely on baking sheets.
To make the icing: Whisk egg white in a glass, metal, or ceramic bowl until foamy. Gradually whisk in confectioners' sugar, a little at a time, until thickened; stir in lemon juice and vanilla extract. Pour icing into a piping bag fitted with a small tip.
Pipe a thin ring of icing onto the base of the largest ring and adhere it to a serving plate. Pipe icing in a zig-zag pattern over the ring. Stack remaining cake rings on top, from largest to smallest, piping icing in a zig-zag pattern between each ring. Set aside to allow icing to set before serving.