Mix the confectioners' sugar and most of the egg whites in a bowl to form a runny paste.
Knead the almond paste to soften, then flatten into a thick disc. Little by little, knead the confectioners' sugar paste into the almond paste until you have a uniform dough. Shape the dough into a ball, then roll into a cylinder and wrap with plastic wrap. Refrigerate for at least 4 hours.
Preheat the oven to 375 degrees F. Line 2 large baking sheets with parchment paper.
Cut the dough into 4 pieces. Roll each piece into a long rope about 3/4-inch thick. Place the pistachios on a large baking sheet and gently roll the ropes in the pistachios to coat.
Gently press the tops of the ropes with the heel of your hand to flatten them slightly--this will make it easier to form them into rings. Run an offset spatula under the ropes to loosen.
Cut the dough into strips (these will become rings): The smallest strip should measure 3 inches; the next, 4 inches; and so on. Continue in 1-inch increments until you reach the largest strip, which should measure 13 inches long.
Shape the strips into rings and push the edges together to seal. Make sure the curved surface is on the outside of the rings. (The smaller rings are the hardest to make, so take your time shaping them.)
Place the rings on the prepared baking sheets. Place a piece of parchment on top of the rings. Place an empty baking sheet on top of the parchment paper and press gently to flatten the tops of the rings. Remove the baking sheet and parchment paper and bake the rings until lightly brown, 10 to 12 minutes.
Once the rings have cooled, begin to build the tower. Dip the base of the largest ring in the melted chocolate and place it, chocolate-side down, in the center of a serving plate. Repeat with the remaining rings, stacking them one on top of the other, from largest to smallest, until the tower is complete. Dust with confectioners' sugar before serving.