Preheat your oven to 350°F (175°C). Prepare an 8-inch round cake pan by spraying it with baking spray or buttering and flouring it. Line the bottom with a circle of parchment paper and apply another layer of spray or butter and flour.
Whisk together the all-purpose flour, baking powder, and kosher salt in a mixing bowl until thoroughly combined. Set aside.
In the bowl of a stand mixer fitted with the whisk attachment, or using a large bowl with a hand mixer, beat the eggs and sugar on medium speed until the mixture is pale, thick, and fluffy, approximately 4 minutes. The mixture should leave a trail when the beater is lifted.
Heat the milk and butter together in a microwave-safe container until the butter is melted and the milk is steaming, using 30-second intervals at high power.
Reduce the mixer speed to low and gradually add the dry ingredient mixture to the egg mixture until just combined. Stir the vanilla into the hot milk mixture, then slowly add this to the batter, continuing to beat until fully combined.
Transfer the batter into the prepared cake pan. Bake in the preheated oven until a toothpick inserted into the center comes out clean, and the cake begins to pull away from the sides of the pan, approximately 33 minutes.
Allow the cake to cool in the pan for a few minutes before inverting it onto a wire rack to cool completely.