Preheat the oven to 350 F. Butter and flour a cake or loaf pan. I used a 9 x 9 inch square baking pan, which worked well for the thickness I wanted, but you could use a round cake pan, or bake it in a loaf pan and cut the cake into cubes.
In a medium bowl, whisk together the flour, baking powder, and salt vigorously to break up the cake flour lumps.
Slice the vanilla bean half lengthwise and scrape out the vanilla seeds with the back of a knife. In a large heatproof bowl, whisk together the melted butter, cream, vanilla seeds and extract until well combined. If the mixture gets too thick as you whisk it, or the butter starts to get lumpy, heat it for a couple of seconds in the microwave.
In a stand mixer fitted with the whisk attachment, whip the eggs and granulated sugar on medium speed until light and fluffy, about 4-5 minutes.
With a rubber spatula, gently fold the egg mixture into the flour mixture until just combined. Fold about 1/4 of this mixture into the butter-cream mixture until just combined. Then fold in the rest of the egg-flour mixture until well-combined.
Pour into the prepared pan. Bake in the preheated oven until golden brown and the center of the cake springs back when pressed in the center. The baking time will vary greatly based on the size and type of pan you used, but in my 9 x 9 inch square pan, it took about 35 minutes. In a loaf pan it will take about an hour.
Cool the cake in the pan for about 30 minutes and turn onto a cooling rack to cool completely.