Line baking trays with parchment paper or silicone baking mats. Preheat oven to 350 degrees Fahrenheit.

In the bowl of a stand mixer with the paddle attachment, cream the butter and sugars. Add the vanilla and eggs. Beat until smooth.

Sift in the flour, baking soda, cornstarch, and cocoa powder. Stir until combined.

Scoop the cookie dough and round into balls (about 1½ Tbsp each). Place on prepared baking sheets, 2-inches apart. Flatten to ½-inch thick with the bottom of a drinking glass dipped in sugar to prevent sticking.

With an electric hand mixer and a large bowl, beat the butter and coconut oil together until smooth. Add the sifted powdered sugar, cocoa powder, and salt. Then add the vanilla and milk. Mix thoroughly. Add extra milk 1 teaspoon at a time to desired consistency.

Spread frosting over cooled cookies and enjoy! Store cookies in a sealed container at room temperature up to 5 days.