Preheat oven to 350° F. Spray 13 x 9-inch baking pan with nonstick cooking spray.
Combine flour, cocoa and salt in large bowl. Combine buttermilk, beet juice and 1 teaspoon vanilla extract in small bowl; whisk to combine.
Beat granulated sugar and 1/2 cup butter in large mixer bowl until smooth and creamy. Add eggs one at a time, beating well after each addition.
Beat in one third of the flour mixture followed by half of the buttermilk mixture until incorporated. Beat in a third of the flour mixture and remaining buttermilk mixture until combined. Beat in remaining flour mixture. Combine vinegar and baking soda in small bowl (mixture will foam); stir into batter. Spoon batter into prepared baking pan.
Bake for 25 to 28 minutes or until wooden pick inserted in center comes out clean. Remove from oven; poke holes all over top of cake with wooden skewer. Combine sweetened condensed milk and evaporated milk in small bowl; stir to combine. Drizzle over top of cake. Cool cake completely before frosting.
Beat cream cheese and remaining 1/4 cup butter in large mixer bowl until smooth and creamy (should take about 4 minutes). Reduce speed to low and gradually add powdered sugar and remaining 2 teaspoons vanilla extract. Increase speed to medium-high and beat until light and fluffy. Spread over top of cake.
Decorate as desired. Store any leftover cake tightly covered in the refrigerator.