Preheat oven to 350°F. Line baking sheets with parchment paper and set aside.
In a large bowl, add melted butter, both sugars, and vanilla and whisk to combine.
Add egg and whisk well until the mixture is glossy and a bit lighter in color, about 1 minute. Let the mixture stand for about 5 minutes while you prepare the next ingredients.
In a medium bowl, add flour, cornstarch, baking soda, and salt and whisk to combine.
Give the sugar mixture a quick whisk and then add the flour mixture. Use a wooden spoon or a spatula to gently stir the flour mixture into the sugar mixture. Stir just until mixed.
Add sprinkles and stir just until blended.
Scoop 1 ½ Tablespoon balls of dough, form each dough ball into a tall ball, and place onto the prepared baking sheets.
Bake the cookies for 8 to 9 minutes. The cookies will still appear pale and puffy when you remove the tray from the oven. Immediately after removing the cookies from the oven, place a round metal biscuit cutter over each cookie and carefully and quickly swirl it around to form the cookie into a smaller perfect circle. Allow cookies to cool on the tray for 5 minutes before transferring to a wire rack to cool completely.
Make the frosting.
In the bowl of a stand mixer using the paddle attachment (or in a large bowl using a handheld electric mixer), beat butter on medium-high speed until smooth.
Add vanilla and salt and beat on medium-high speed until blended.
Add sugar slowly – one cup at a time – beating well on medium speed and scraping the bowl often. When all sugar is added the frosting will be slightly dry.
Add milk and beat on high speed until the frosting is light and fluffy.
Decorate the cookies.
Once the cookies are completely cooled, add frosting to a piping bag fitted with a large round tip and pipe a swirl of frosting on top of each cookie. Alternatively you can simple spoon a bit of frosting onto each cookie.
Use a small offset spatula to spread the frosting around the cookie, creating an edge of frosting that will help to keep the sprinkles contained.
Add sprinkles onto frosting and gently push them down with your fingers to ensure they stick in the frosting.
Notes
Storage: These cookies stay fresh for up to 3 days when stored in an airtight container at room temperature. Freezing: Dough can also be frozen and baked at a later date or unfrosted cookies can be baked and stored in an airtight container in the freezer for up to 2 months. Check out my tips for freezing cookie dough and baked cookies.To help ensure the cookies are thick, form the dough into tall balls before baking. Immediately after removing the cookies from the oven, place a round metal biscuit cutter over each cookie and carefully and quickly swirl it around to form the cookie into a smaller perfect circle.