For the cookies: Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Line 2 large (18 by 12-inch) baking sheets with parchment paper.
Sift flour, cocoa, baking soda, and salt together into medium bowl. With electric, beat butter and sugar at medium speed until light and fluffy, about 3 minutes. Scrape down sides of bowl with rubber spatula. Add cooled espresso and vanilla and beat at medium-low speed until combined. Scrape down bowl. With mixer on low speed, gradually add dry ingredients and mix until just combined.
Roll dough into thirty 1-inch balls. Place 15 balls on each prepared baking sheet, spacing them about 2 inches apart. Bake until just set, rotating sheets from front to back and top to bottom halfway through baking, 8 to 9 minutes. Cool cookies on sheets for 2 minutes; transfer with metal spatula to wire rack to cool completely, at least 30 minutes. (Cookies can be stored in airtight container for up to 4 days.)
For the frosting: With an electric mixer, beat butter at medium-high speed until smooth, about 20 seconds. Add confectioners' sugar and beat at medium-low speed until most of the sugar is moistened, about 45 seconds. Scrape down bowl and beat at medium speed until mixture is fully combined, about 15 seconds. Scrape down bowl. Add milk, peppermint extract, and food coloring and beat at medium speed until incorporated, about 10 seconds, then increase speed to medium-high and beat until light and fluffy, about 4 minutes, scraping down bowl once or twice. Frost cooled cookies and immediately sprinkle each with crushed mints. (Once frosted, cookies are best served within 1 day.).