Separate the eggs by placing the egg whites in a clean mixing bowl and the egg yolks in another medium bowl.
Beat the egg whites until medium peaks form.
Whisk the buttermilk, butter, and vanilla extract into the egg yolks until smooth.
In a large bowl, mix together flour, sugar, baking powder, and salt. Stir in the egg yolk mixture until just combined. Gently fold in the beaten egg whites just until no white streaks remain.
Place the aebleskiver pan over medium-low heat. Use a pastry brush to grease each hole with oil. If desired, place the nutella in a piping bag or ziploc bag with the corner snipped off.
Once heated, quickly fill each hole barely halfway with the batter. Pipe or spoon about 1 teaspoon of nutella into the center of each aebleskiver. Cover with more batter, until each hole is almost filled.
Cook until the edges are set and golden, adjusting the heat as needed to about medium low, about 2 minutes.
Use a small spoon, skewer, or aebleskiver turner to flip the aebleskiver 1/3 of the way over.
Cook another minute before flipping the rest of the way over to cook the remaining batter. Use the small spoon, skewer, or aebleskiver tool to transfer to a plate. Grease as needed and repeat with remaining batter and nutella.
Serve warm with powdered sugar or maple syrup.