Preheat the oven 375 degrees F. Line several baking sheets with parchment paper and set aside.
Cream the butter and sugar together with an electric mixer until light and fluffy, 3-5 minutes. Scrape the bowl with a spatula, and turn the mixer on low. Beat in the salt, eggs, vanilla, and then finally the flour. Once combined, turn the mixer off as to not over-mix. Wrap the dough in plastic wrap and chill for 30 minutes.
Once the dough is cold and firm, flour a work surface. Cut the dough in half and roll it out into an even 1/8-inch sheet. Using 4 sizes of star cutters (4-inch, 3-inch, 2-inch, and 1-inch) cut two stars out with each cutter, for each tree, followed by one 1 1/2-inch circle for the base and seven half-circles for the connectors. You should have a total of 8 stars, 7 small connector circles, and one medium base circle per tree. Repeat with remaining dough.
Place the cookies on the baking sheets 1 1/2 inches apart. Bake for 6-7 minutes, until the edges are just barely golden. Cool completely on the baking sheets.
Once the cookies are cool, whisk together 2 cups of powdered sugar and 3 tablespoons half & half. The mixture should have the consistency of school glue. If it's too thick, add a little more half & half.
Use the icing to glue the 1 1/2-inch circles to the bottoms of the largest stars. Then glue a small round "connector" cookie in the middle of the stars and add the second large star on top, shifting the cookie so the points are offset. Continue to stack and shift the points until you have eight star cookies stacked from largest to smallest, to form a tree. Allow them to dry before decorating. (See images above for guidance.)
To decorate: Drizzle the remaining icing over the trees and sprinkle with sugar sprinkles. You could also dip larger candies into the icing and set them on the points. Dry completely before wrapping as gifts.
NOTES: 1) Make sure to cut the cookies close together. The more you gather and reroll the dough, the tougher the cookies will be. 2) You can cut extra stars and connector circles to make some of the trees a little taller.