Line a large rimmed baking sheet with parchment paper. Place the granulated sugar in a small bowl and set aside for coating the mints.
In a large mixing bowl, combine the softened cream cheese, unsalted butter, light corn syrup, and peppermint extract. Using a handheld mixer on low speed, beat for approximately 1 minute, or until smooth and creamy.
Gradually add 4¼ cups powdered sugar, about ½ cup at a time, mixing on low speed after each addition. As the mixture thickens into a stiff dough, switch to a spatula or wooden spoon if needed.
Place a large sheet of parchment paper on a clean work surface and sprinkle it with ¼ cup of the remaining powdered sugar. Transfer the peppermint dough onto the prepared surface.
Divide the dough into three equal portions. Make a small indentation in the center of two portions. Add 6 drops red food coloring to the first portion and 6 drops green food coloring to the second. Leave the third portion white.
Knead the red dough until the color is evenly distributed. If sticky, gradually work in up to ¼ cup additional powdered sugar. Repeat with the green dough. If needed, knead 1–2 tablespoons powdered sugar into the white dough to achieve a similar consistency.
Roll each colored portion of dough into a long rope measuring approximately 18 inches in length and 1 inch in thickness.
Using a knife, slice each rope into ½-inch pieces. Repeat with all three dough ropes.
Roll each piece of peppermint dough between your palms to form a small, smooth ball.
Place 2–3 dough balls at a time into the bowl of granulated sugar and gently toss until evenly coated. Transfer to the prepared baking sheet, leaving a little space between the candies.
Using the tines of a fork, gently press down on each sugar-coated ball to flatten it into a small, thick disc with a decorative striped pattern.
Allow the Christmas Mints to set until firm enough to handle. The original recipe specifies uncovered room-temperature drying for up to 24 hours, but because the mints contain cream cheese, do not leave them unrefrigerated that long. Refrigerate for extended setting and limit total time at room temperature to 2 hours.
Once the mints have set, transfer them to a serving platter and enjoy.