In a large bowl, cream the softened butter with both sugars until light and fluffy.
Add eggs one at a time, beating well after each. Stir in vanilla extract.
In a separate bowl, whisk together flour, black cocoa powder, cornstarch, baking soda, and salt.
Gradually add dry ingredients to the wet ingredients. Mix until just combined.
Fold in the crushed Oreos gently. Chill dough for at least 30 minutes.
Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Scoop dough into ¼ cup portions, spacing well on the baking sheets. Bake for 12–15 minutes until edges are set.
Let cookies cool on the baking sheet. Prepare the frosting while they cool completely.
Beat softened butter for the frosting until smooth. Gradually add powdered sugar, milk/cream, and vanilla. Mix until fluffy.
Divide frosting into three bowls. Add lemon extract and yellow coloring to one, berry flavoring and pink/purple to another, and blue to the last. Mix each color thoroughly.
Once cookies are cold, dollop a bit of each frosting color on top. Gently swirl using the back of a spoon or offset spatula.
Drizzle with melted white chocolate and sprinkle with crushed Oreos.
