Line baking sheets with parchment paper or silicone baking mat and set aside. Sift flour, cornstarch, baking powder, baking soda, and salt into a bowl.
In a stand mixer fitted with the paddle attachment beat 1/2 cup butter, sugar, and vanilla until fluffy. Add egg yolks one at a time until fully incorporated.
Alternate between adding flour and peppermint candy to wet mixture while beating on medium-low speed. If dough appears dry, add cream 1 tablespoon at a time until dough is soft. Divide into two pieces, cover, and chill for 1 hour.
Preheat oven to 325F. While dough is chilling, make the filling. Using the whisk attachment on a stand mixer with a clean bowl whip remaining butter, 2 tablespoons cream, and peppermint extract on medium-low speed.
Once cream is fluffy, add red food coloring one drop at a time until desired red is achieved. Chill until ready to use.
Working with one piece of dough at a time, roll to 1/4-inch thick on a well floured surface. Using a 2 1/2-inch circular cookie cutter, cut rounds and transfer to baking sheets. Repeat with remaining dough.
Bake cookies for 15 minutes. Transfer cookies to cooling rack and cool completely.
Pair each cookie with another that is a similar shape and size. Add 1 tablespoon filling to the flat side of each cookie, then sandwich together. Press cookies together until cream begins to pillow out of the sides.
Smear a bit of cream on cookies and top with a little piece of peppermint candy.