Beat the butter, superfine sugar and salt in a large bowl with a mixer on medium speed until light and fluffy, about 3 minutes. Beat in the egg, vanilla and peppermint extract. Reduce the mixer speed to low and beat in the flour until incorporated. Transfer the dough to a piece of plastic wrap and pat into a rectangle; tightly wrap and refrigerate until firm, at least 1 hour or overnight.
Line 2 baking sheets with parchment paper. Spread some red sanding sugar and superfine sugar on separate plates. Gently roll heaping teaspoonfuls of dough into short logs with your hands. One at a time, roll half of the logs in red sanding sugar to form 4-inch strands. Repeat with the remaining logs and superfine sugar. Twist 1 strand of each color together, then arrange on the prepared baking sheets, about 2 inches apart, and bend the tops to form candy canes. Refrigerate until firm, about 15 minutes. Meanwhile, position racks in the upper and lower thirds of the oven and preheat to 350 degrees F.
Bake, switching the pans halfway through, until the cookies are set and lightly browned around the edges, 11 to 13 minutes. Let cool completely on the baking sheets.