Preheat the oven to 350°F. Grease and line a 6-inch round cake pan.
In a medium bowl, whisk together the flour, granulated sugar, cocoa powder, baking powder, baking soda, and salt.
In another bowl, stir the beet puree, vegetable oil, egg, buttermilk, and vanilla extract until smooth.
Pour the wet mixture into the dry ingredients and stir gently until just combined. Do not overmix.
Transfer the batter to the prepared pan and bake for 30 to 35 minutes, or until a toothpick inserted in the center comes out clean. Cool completely in the pan.
In a mixing bowl, beat the softened butter until creamy. Gradually add the powdered sugar and pomegranate molasses, beating until the frosting is light and spreadable.
Remove the cooled cake from the pan. Spread the buttercream over the top and sides. Scatter the pomegranate seeds over the surface before serving.