In a small bowl, blend cocoa powder and oil until smooth. Set aside. Heat oven to 350 degrees F. Place liners in two standard-size muffin pans, for a total of 24 cupcakes. In a medium-size bowl, whisk flour, baking powder and salt.
With electric mixer, beat butter and sugar in large bowl until light-colored and smooth. Add eggs, one at a time, beating well after each addition. On low speed, add flour mixture, alternating with milk. Stir in vanilla. Divide batter in half. Stir cocoa mixture into one half. Spoon both batters into prepared liners, 1/3 cup in each (12 chocolate, 12 vanilla).
Bake at 350 degrees F for 25 minutes or until firm to the touch. Let cupcakes cool in pan on rack for 5 minutes, then transfer cakes directly to rack; cool completely.
Beat butter, confectioners' sugar, milk and vanilla on low speed to blend, then on high speed for 2 minutes, until good spreading consistency. Divide frosting into bowls and tint to desired colors. Transfer frosting to plastic bags and snip off one corner. Pipe in swirled pattern onto cooled cupcakes.