Preheat the oven to 350°F (175°C). Grease two 8-inch round pans and line the bottoms with parchment.
Whisk together the flour, baking powder, baking soda, and salt.
Combine the water, sugar, butter, vegetable oil, vanilla extract, and vanilla bean or paste in a saucepan.
Warm over medium heat until the butter melts and the sugar begins to dissolve. Do not boil hard.
Cool the mixture for about 10 minutes.
Whisk the eggs separately, then slowly whisk them into the warm vanilla mixture.
Pour the wet mixture into the dry ingredients and whisk until smooth. The batter will be thin.
Divide evenly between the pans.
Bake for 25–30 minutes, until lightly golden and a toothpick inserted into the center comes out clean.
Cool in the pans for 10–15 minutes, then transfer to wire racks and cool completely.
Beat the room-temperature butter until smooth.
Gradually beat in the powdered sugar.
Add the vanilla and 2 tablespoons cream, half-and-half, or milk. Add the final tablespoon only if needed.
Spread frosting over the first cake layer, add the second layer, then frost the top and sides.
Decorate with sprinkles if desired.
