Beat together the butter and confectioners' sugar in the bowl of a stand mixer until combined. Add the vanilla and mix until just combined.
In a separate large bowl, whisk together the flour, ground pistachios and salt. Gradually add the flour mixture to the butter and sugar, beating just to combine. Add the dried cherries and mix just until combined; the mixture will look crumbly. Turn the dough out onto a clean work surface and knead few times with your hands until the dough comes together. Divide into 3 pieces and form each piece into a disk; wrap with plastic wrap and chill for at least 1 hour.
When ready to bake, preheat the oven to 325 degrees F. Line 4 baking sheets with parchment paper.
Roll the dough, one disc at a time, to 1/3-to-1/2-inch thick. Cut 2 rounds with a 6-inch cutter. Cut the center from each with a 4-inch cutter to create rings. Then cut 2 rounds with a 5 1/2-inch cutter and use a 3 1/2-inch cutter to create rings. Cut 2 rounds with a 5-inch cutter and use a 3-inch cutter to create rings. Cut 2 rounds with a 4 1/2-inch cutter and use a 2 1/2-inch cutter to make rings. Cut 2 rounds with a 4-inch cutter and use a 2-inch cutter to make rings. Cut 2 rounds with a 3 1/2-inch cutter and use a 1 1/2-inch cutter to make rings. Cut 2 rounds with a 3-inch cutter and use a 1-inch cutter to make rings. Cut 2 rounds with a 2 1/2-inch cutter and 2 rounds with a 2-inch cutter, leaving them whole. You may gather and reroll the scraps up to 2 times if needed but you will need to chill the dough before rerolling.
Arrange the cookies on the prepared baking sheets, leaving at least 1 inch of space in between. Bake until the edges of the cookies are lightly golden brown, about 22 minutes. Remove from the oven and cool the cookies completely on a wire rack before assembling.
When ready to assemble, stack the two 6-inch ring cookies atop each other, using the white Royal Icing to adhere to one another creating a sandwich. Repeat with the other like-size cookies so they are all in 2-cookie stacks.
To assemble the tower, pipe white icing on the bottom of the stack of 6-inch cookies and set it on a decorative platter (the icing will help it adhere). Decorate the cookies with swags of red icing and add the next-largest sandwich. Continue decorating with swags of alternating colors of icing and stacks of cookies in descending sizes. Finish with the solid rounds on top.
Add the confectioners' sugar and meringue powder to the bowl of a stand mixer fitted with the whisk attachment; turn on low to combine. Add the water 1 tablespoon at a time, beating with the mixer on medium speed until the icing is thick and glossy but soft enough to pipe. Divide among 3 bowls, tinting one red, one green and leaving one white. Transfer the icing to plastic piping bags; snip off the corners.
If you don't have cookie cutters in these sizes, check around your kitchen for bowls and glasses in various sizes and cut around those with a paring knife.