In a large mixing bowl, beat butter and sugar with an electric mixer until light and fluffy.
Add eggs, one at a time, beating after each addition.
Add peppermint extract then mix until combined. Set aside.
In a separate bowl, whisk together flour, cornstarch and salt.
Gradually add dry ingredients to wet, mixing on low speed until just combined. Do not overmix.
Remove half of dough from the mixing bowl and place between 2 sheets of parchment paper. Roll the dough into a 14 x 12 inch rectangle that is 1/4 inch thick. Keep dough between the parchment layers and place in the refrigerator to chill for 30 minutes.
Add ground candy canes and red food coloring to the dough in the mixing bowl. Beat on low until combined then place dough between 2 sheets of parchment paper. Roll the dough into a 14 x 12 inch rectangle that is 1/4 inch thick. Keep dough between the parchment layers and place it in the refrigerator to chill for 30 minutes.
After dough has chilled, place the plain dough on your work surface and very gently remove the top layer of parchment only. Next, gently remove both sides of parchment from the candy cane dough and place on top of plain dough. Trim any uneven edges with a pizza cutter.
Use a rolling pin to gently seal the two layers together. It should only take 1 or 2, light-pressure rolls on the peppermint layer to seal. Be careful not to flatten dough.
Starting with the long end, gently roll the two layers together into a log. Tugging on the parchment helps tighten the roll. The dough may be slightly sticky, so take your time with the process.
Cut the log into 2 halves. Wrap each in parchment and refrigerate for 8 hours.
After dough had chilled, preheat oven to 375 degrees. Line baking sheets with parchment paper and set aside.
Unwrap one log of dough and slice into 1/4 inch rounds. (Fishing line is my trick to making clean cuts, but you can also use a sharp knife.)
Place pinwheels on baking sheet, spaced 1 inch apart and bake for 8 minutes. Note, cookies will look puffy when they come out of the oven but will flatten as they cool.
Cool on baking sheets for 5 minutes then transfer to a baking rack to cool completely.
Repeat the slicing and baking process for the remaining log of dough.
Store cookies in an airtight container at room temperature up to 5 days.