Preheat the oven to 325°F, and adjust a rack to the center.
Add the milk to a small saucepan, and use 1 teaspoon of butter to grease the sides of an (approximately 9 X 9 X 3-inch) cake pan -- square or round. Line the bottom of the pan with parchment paper, and set aside.
Use a paring knife to slice open the vanilla bean, lengthwise. With the back of the knife, scrape the vanilla into the saucepan with the milk and add the emptied pod as well.
Place the saucepan over low-medium heat, and bring it to a simmer. Almost as soon as it begins to simmer, turn the heat off and let it sit for about 10 minutes. Then remove the pods and add the vanilla extract. Stir to blend. Set aside.
In a mixing bowl, mix the flour with the baking powder and salt. set aside.
In another (larger) mixing bowl, and using an electric mixer (or strong arm and a spoon), mix butter and the sugar until combined. Add the eggs, one at a time, mixing well after each addition. Add the lemon zest and lemon juice and mix just to blend.
While continuing to mix, slowly add dry ingredients until combined.
Then, still mixing, drizzle in vanilla-milk and mix until a smooth batter forms.
Pour batter into the prepared cake pan and bake until a toothpick inserted into center of cake comes out clean about 1 hour.
Let the cake cool slightly, then invert onto a wire rack and let cool completely. Then invert it again onto a cake plate.