To prepare cake, place beet in a small saucepan; cover with water. Bring to a boil. Cook 10 minutes or until tender. Drain and cool slightly.
Place beet in a food processor; process until very finely chopped. Remove 1 cup beet from processor; reserve remaining beet for another use. Combine 1 cup beet, buttermilk, and 1 1/2 teaspoons vanilla in food processor. Process until very smooth.
Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, cocoa, baking powder, and 1/2 teaspoon salt in a bowl, stirring well with a whisk.
Place 1 1/2 cups granulated sugar, butter, and oil in a large bowl; beat with a mixer at medium speed until fluffy (about 5 minutes). Add eggs and 1 egg white, 1 at a time, beating until incorporated. Add flour mixture and beet mixture alternately to butter mixture, beginning and ending with flour mixture. Divide batter evenly among 3 (8-inch) round metal cake pans coated with baking spray. Bake at 350° for 23 to 24 minutes or until a wooden pick inserted in center comes out clean. Cool 15 minutes in pans on a wire rack. Remove cake from pans; cool completely on wire rack.
To prepare filling, place cream cheese, cream, yogurt, 1/2 teaspoon vanilla, and dash of salt in a medium bowl; beat with a mixer at high speed until smooth. Add powdered sugar; beat at low speed 1 minute or until well combined (do not overbeat).
To prepare meringue, combine 2/3 cup granulated sugar and remaining ingredients in the top of a double boiler. Cook over simmering water 2 to 3 minutes or until a candy thermometer registers 160°, stirring constantly with a whisk. Remove from heat. Beat egg mixture with a mixer at medium speed until soft peaks form; beat at high speed until stiff peaks form (do not overbeat).
Place 1 cake layer on a plate, and spread half of filling over top; top with another cake layer. Spread remaining filling over top; top with remaining cake layer. Spread meringue frosting over top and sides of cake.