In a blender, combine the milk, flour, eggs, sugar, vegetable oil, food coloring and cocoa until smooth. Let rest 30 minutes.
Preheat a 9-inch skillet over medium low heat. Lightly grease with butter. Add ¼ cup crepe batter and use your wrist to swirl the batter around until it completely covers the bottom of the pan. After 1 minute, use a rubber spatula to loosen the edges of the crepe. Use the spatula or your fingers to flip the crepe and let it cook for 1 more minute. Slide crepe out of the pan and repeat. Let crepes cool completely.
To make the filling, beat the cream cheese and butter until smooth. Add the powdered sugar and vanilla extract and mix on low until combined, then increase speed to medium high and beat until smooth and light.
To assemble the crepe cake, place 1 crepe on a cake stand or large plate. Cover with a few tablespoons of the filling, spreading it very thin. Repeat with remaining crepes and filling. Chill in the fridge for 30 minutes to firm up.
Dust with powdered sugar before serving. Store leftovers in the fridge.