Optional sugar coating: Stir granulated sugar and white sparkling sugar together in a small bowl (optional but gives the cookie a nice crunchy exterior and additional sweetness).
Place the melted butter in a large bowl and allow it to cool to room temperature before adding sugar. (If the butter is hot, it will melt the sugar and cause greasy cookies.) Add granulated sugar and stir with a wooden spoon until smooth and incorporated, about 3–4 minutes. Add the whole egg, egg yolk, vanilla extract, and peppermint extract. Stir until smooth.
Add the peppermint crunch sprinkles, flour, salt, cornstarch, baking powder, and baking soda to the wet ingredients. Mix until just combined. Do not overmix the dough.
Roll tall balls of dough, each using a full 2 tablespoons of dough. If you have a kitchen scale, the balls of dough should be about 43 grams. Roll the dough balls generously in the granulated/white sparkling sugar mixture. Cover and refrigerate dough balls for 1 hour and up to 8 hours.
Preheat oven to 325°F. Place dough balls on a parchment or silicone-lined sheet pan, spread far apart (I only bake 6 cookies at a time; they spread a lot), and bake for 9–14 minutes. Watch carefully, being sure to not overbake. (Slightly underbaked sugar cookies are the best! I pull them out at 11–12 minutes.) Remove and immediately press the edges of the cookie toward the center with the back of a spoon or metal spatula. Let stand on sheet pan for 2 minutes before removing to a cooling rack with a spatula.
Frosting: Prepare frosting while the cookies bake. In a large bowl, beat together the softened butter and cream cheese until completely smooth. Add vanilla, peppermint extract, and salt. Beat to combine. Add powdered sugar and beat until frosting is smooth (it will seem like you need more liquid, but just keep beating; it will come together!)
Frost each completely cooled cookie and sprinkle with crushed peppermint. Store any leftover cookies in an airtight container in the fridge. Cookies are best enjoyed within 2–3 days.
Notes
Note 1: Use Peppermint Crunch Sprinkles® or Candy Cane Crunch®. If you can't find the crunch/sprinkles, use 1/4 cup crushed up candy canes instead (blend in a blender or pulse in a food processor).Note 2: If you press a measuring cup into a bag of flour, you will pack in way too much flour, resulting in the wrong texture of cookie. To accurately measure the flour, spoon flour into the measuring cup until it’s overfilled. Then use the back of a table knife to level the measuring cup at the top. (Visual here).Storage: Keep leftover cookies in an airtight container in the fridge. They are best enjoyed within 2–3 days.