In a large bowl, cream the butter and sugar until light and fluffy. Add eggs, one at a time, beating well after each egg. Beat in bananas and vanilla. Combine the flour, baking powder, and baking soda; add to creamed mixture alternately with milk, beating well after each addition.
Transfer to a greased 9×13 inch baking pan. Bake at 350 degrees for 25-30 minutes (or until toothpick inserted in the middle comes out clean). Cool on a wire rack.
For frosting, in a large bowl, beat cream cheese and butter until light and fluffy. Beat in vanilla. Gradually add powdered sugar until smooth. Spread over bars. Store covered in the fridge.