Foryoueats: Doberge Cake (Dobash)

1 (15.25 ounce) package white cake mix (such as Duncan Hines)

1 tablespoon all-purpose flour

1 tablespoon white sugar

1 teaspoon baking powder

1 ⅓ cups water

3 egg whites

2 tablespoons vegetable oil

2 cups milk

1 (3.9 ounce) package instant chocolate pudding mix

1 (4 ounce) package cream cheese, softened

¼ cup unsalted butter, softened

1 (16 ounce) box confectioners' sugar

½ cup unsweetened cocoa powder

½ teaspoon vanilla extract

1 tablespoon milk, or as needed

Doberge Cake (Dobash)