Preheat the oven to 350°F (175°C) and line a large baking sheet with parchment paper or a silicone baking mat.
In a large mixing bowl, combine the chocolate cake mix, eggs, and vegetable oil; stir until a thick, sticky dough forms with no dry mix remaining.
Use a cookie scoop or spoon to drop rounded balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake the cookies for 8 to 10 minutes until edges are set but centers remain slightly soft; avoid overbaking.
While cookies bake, melt butter in a medium saucepan over medium heat.
Whisk in cocoa powder and milk into the melted butter until smooth, then remove from heat.
Stir in vanilla extract and powdered sugar; whisk vigorously until frosting is smooth and glossy. Adjust consistency with milk or powdered sugar as needed.
Let cookies cool for 2-3 minutes on the baking sheet, then spoon about a tablespoon of warm frosting onto each cookie, gently spreading toward the edges.
Transfer frosted cookies to a wire rack and allow them to cool completely so the frosting sets and hardens slightly before serving.