Preheat your oven to 350°F. Grease an 8-inch round cake pan with baking spray or butter, then line the bottom with parchment paper and grease it again.
In a small bowl, whisk together 1 cup flour, 1 ¼ teaspoons baking powder, and ¼ teaspoon salt, then set aside.
Using a stand mixer fitted with a whisk attachment (or a large mixing bowl with a hand mixer), beat 2 eggs and 1 cup sugar on medium speed until the mixture is pale, thick, and fluffy (about 4 minutes). When you lift the beater, it should leave a visible trail in the batter.
Meanwhile, in a small saucepan, heat ½ cup milk and 5 tablespoons butter over medium heat until the butter is melted and the milk is steaming.
With the mixer on low speed, gradually add the flour mixture to the egg mixture, beating just until combined. Stir in 2 teaspoons vanilla extract into the hot milk mixture, then slowly pour it into the batter while mixing. The batter should be smooth, slightly thick, and bubbly on top. Scrape down the sides of the bowl and fold a few times to ensure everything is fully combined.
Pour the batter into the prepared cake pan and bake for about 33 minutes, or until the cake is golden brown, a toothpick inserted into the center comes out clean, and the cake begins to pull away from the sides of the pan.
Let the cake cool in the pan for a few minutes before inverting onto a wire rack to cool completely.