OPTIONAL SUGAR COATING: Stir the white sugar and white sparkling sugar together in a small bowl (optional but gives the cookie a nice crunchy exterior and additional sweetness).
WET INGREDIENTS: 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 in the white sugar and stir with a wooden spoon until smooth and incorporated, about 3-4 minutes. Add in 1 whole egg and 1 egg yolk, vanilla extract, and peppermint extract. Stir until smooth.
DRY INGREDIENTS: 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.
CHILL: 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 balls of dough generously in the white sugar/white sparkling sugar mixture. Cover and refrigerate the dough balls for 1 hour, and up to 8 hours.
BAKE: Preheat the oven to 325 degrees 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 over bake. (Slightly underbaked sugar cookies are the best! I pull them out at 11-12 minutes.) Remove and immediately press the edges of the cookie towards the center with the back of a spoon or metal spatula. Let stand on cookie sheet for 2 minutes before removing to a cooling rack with a spatula.
FROSTING: Prepare the frosting while the cookies bake. In a large bowl, beat together the softened butter and cream cheese until completely smooth. Add in the vanilla, peppermint extract, and salt. Beat to combine. Add in the powdered sugar and then beat until the frosting is smooth (it will seem like you need more liquid, but just keep beating; it will come together!)
FROST COOKIES: 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.