In a large bowl, cream together the softened butter, heavy cream, and vanilla extract until pale and fluffy, about 2 minutes.
Gradually add the sifted powdered sugar, one cup at a time, mixing on low speed until fully incorporated and dough is smooth.
Stir in peppermint extract and gel food coloring, if using, until the color is even throughout.
Turn the dough out onto a sugar-dusted surface and roll it into a long rope about 1/2-inch in diameter.
Wrap the rope tightly in plastic wrap and chill in the refrigerator for 30 minutes.
Unwrap the chilled rope and slice it into 1/4-inch thick pieces using a sharp knife or unflavored floss.
Place the slices on parchment-lined trays and let them rest at room temperature for at least 2 hours until the surfaces dry to a matte finish.
Notes
Use room-temperature butter for smooth dough.If the dough feels too dry, add an extra teaspoon of cream.Store mints in an airtight container at room temperature for up to one week.