Make cookies: Whisk together flour, cocoa powder and salt. In a separate large bowl, using an electric mixer on medium speed, beat butter and sugar until fluffy. Beat in eggs, one at a time, then vanilla. Reduce speed to low and gradually add flour mixture; mix until just combined. Divide dough in half. Form each half into a disk and wrap in plastic wrap. Refrigerate until firm, about 1 hour.
Preheat oven to 350°F. Working with 1 disk at a time, roll out dough to 1/4-inch thickness on a lightly floured sheet of parchment. Brush off excess flour, transfer parchment to a baking sheet and freeze dough until firm, about 15 minutes.
Use cookie cutters to create 28 cookies of each size: 21/4", 13/4" and 1". Reroll scraps as needed. Transfer cookies to new parchment-lined baking sheets. Bake until cooked through, 12 to 14 minutes. Let cookies cool completely on sheets.
Make royal icing: Add water 1 Tbsp. at a time, until icing is the consistency of honey. Fill a piping bag with icing. (Or spoon into a ziplock bag; using scissors, cut a small hole in corner.) Cover top of each cookie with icing. Slice small triangular pieces of dried mango for snowman noses and press on center of smallest iced cookies. Slice off tiny pieces of licorice laces to create eyes and buttons; place on iced cookies.
Let icing set, about 2 hours. Add a dab of icing on bottom of all 1-inch and 1 3/4-inch cookies; stack 1 cookie of each size to make snowman. Using a food-safe pen, draw a smile below nose. Repeat with remaining cookies.
GIVE THEM Place snowman cookies in a box lined with tissue or parchment paper.