Place butter and salt in the bowl of a stand mixer. Beat at medium-low speed until smooth and creamy, about 1 minute.
Add 3 cups sugar, half-and-half, vanilla, and mint extract; beat at low speed until a smooth dough forms, 1 to 2 minutes.
With mixer on low speed, gradually add remaining 1 cup sugar, beating until dough is stiff and smooth, 1 to 2 minutes.
Turn dough out onto a work surface dusted with powdered sugar. Knead a few times until smooth.
If desired, divide dough into 2 or 3 equal portions. Add 1 to 2 drops of food coloring to each portion and knead until evenly combined.
Pinch off a lime-sized piece of dough; roll into a ½-inch wide rope on work surface.
Using a bench scraper or knife, cut rope into ½-inch-long pieces. Repeat process with remaining dough.
Arrange cut dough pieces in a single layer on a sheet pan lined with parchment paper.
Let stand, uncovered, at room temperature until completely dry, 8 to 12 hours.
Store in an airtight container with wax or parchment paper between layers for up to 2 weeks.
