To make cookies, in large bowl, mix flour, baking powder and salt. Set aside.
In another large bowl, beat butter and both sugars with electric mixer at medium speed or with spoon until well blended. Beat in eggs, one at a time, scraping bowl after each. Beat in vanilla. Stir in flour mixture until well mixed.
Wrap dough tightly in plastic wrap; refrigerate 2 hours.
Roll out dough on lightly floured surface to 1/4 inch thick. For each cookie cake, cut 3 (3-inch) circles, 2 (2-inch) circles, and 2 (1-inch) circles. Place 2 inches apart on ungreased cookie sheet. Refrigerate 30 minutes.
Heat oven to 350°F. Bake 8 to 10 minutes or until cookies are set. Cool completely, about 15 minutes.
To make icing, in large bowl, beat icing ingredients with electric mixer on medium speed 7 to 10 minutes or until no longer glossy.
Add more warm water to icing, 1/2 tablespoon at a time, until icing can be piped through a tip. Icing will still be stiff, but if too thick to pipe, keep adding water. Place in piping bag fitted with writing tip. Pipe thin line around outside edge of each cookie. Allow to set 15 minutes.
Set aside 1/2 cup of the icing in airtight container. Add more water to remaining icing. Make it thin enough so if you drop a spoonful back into the bowl, the icing does not hold its shape, but disappears back into the frosting within a few seconds.
Drop a spoonful of the icing onto center of each cookie. Using an offset spatula, spread icing to the border piped on earlier. Allow cookies to dry at least 6 hours.
To assemble cookie cake, place reserved icing in piping bag fitted with round tip. Place one of the largest cookies on the bottom, and place a dot of icing in center of cookie as glue. Place a second large cookie on top. Continue the process, using the icing to “glue” the cookies together. For each cookie cake, use 3 large cookies on the bottom, followed by 2 medium-sized cookies and 2 small cookies on top. Place small dot of icing on top cookie, and place small piece of cotton candy on top for decoration. Repeat with remaining cookies to make 8 cookie cakes.