Preheat the oven according to the temperature listed on the cake mix box. Grease a 9×13-inch baking pan.
Beat together the cake mix, milk, sour cream, melted butter, eggs, and vanilla until smooth. Add blue food coloring a little at a time until you reach a bright Cookie Monster blue.
Toss the mini chocolate chips with the flour, then fold them into the batter. Spread the batter evenly in the prepared pan and bake according to the time listed on the cake mix box, or until a toothpick inserted into the center comes out clean. Let the cake cool completely.
To make the cookie dough buttercream, beat the butter until light and fluffy. Gradually beat in the powdered sugar, heat-treated flour, vanilla, and salt. Add the heavy cream a tablespoon at a time until the frosting is smooth and fluffy. Fold in the mini chocolate chips.
Spread the buttercream over the cooled cake. Top with mini chocolate chip cookies, extra mini chocolate chips, and a mix of blue and chocolate sprinkles.