Preheat oven to 350°F. Line 2 baking sheets with parchment paper and set aside.
In a large mixing bowl beat butter and cream cheese together with an electric mixer until well blended. Add 1 cup of sugar and beat until light and fluffy. Add the egg, vanilla and peppermint extracts. Beat until combined.
In a small bowl whisk together the flour and salt. Add half the flour mixture to the butter and sugar and blend on low until incorporated. Add the remaining flour and beat until a soft dough forms.
Transfer half the dough to a clean bowl. (A scale is helpful to ensure it’s divided equally.) Add the red food coloring and blend until no streaks remain.
Pour the remaining ¾ cup of granulated sugar into a pie plate.
Lightly flour a clean work surface. Scoop out 1 teaspoon of the uncolored dough and roll into a 4-inch rope. Scoop out 1 teaspoon of the red dough and also roll into a 4-inch rope. Lay the ropes side-by-side and gently roll the two ropes together to adhere, taking care to keep the length at about 4-inches.
Transfer the rope to the granulated sugar and roll to coat. Twist the rope a few times and transfer to the baking sheet. Form into a candy cane by curving one end into a hook. Repeat with remaining dough spacing the cookies 1-inch apart.
Refrigerate the unbaked cookies for 30 minutes before baking.
Bake the cookies for 10-12 minutes or until firm to the touch and just starting to brown around the edges. For best results, do not over-bake. Cool on the pan for 5 minutes then remove to a wire rack and cool completely. (Optional: while the cookies are hot from the oven, sprinkle with additional sugar, if desired. The sugar will melt on the hot cookie and adhere.)