Combine butter and 2 cups sugar in the bowl of a stand mixer fitted with the paddle attachment. Mix on low speed until sugar is moistened, then increase speed to medium-high and beat until well combined. Return mixer to low speed, add remaining 2 cups sugar, and mix until well combined, 1 minute. Increase speed to medium-high; beat until light and fluffy, 6 minutes more. Add peppermint extract, milk, and salt; beat until smooth.

Lightly dust a clean work surface with confectioners' sugar and turn out "dough" onto it. Divide into 5 pieces; tint as desired with food coloring, leaving one piece white. (Keep covered with plastic wrap while working to prevent drying.)

Roll "dough" into 1/2-inch-thick ropes, then cut into pieces slightly over 1/2 inch long with a paring knife or bench scraper. Transfer to a parchment-lined rimmed baking sheet. Let dry, uncovered, at least 12 hours. Mints can be refrigerated in an airtight container up to 2 weeks.