Preheat oven to 350 degrees F (175 degrees C). Grease a 9 1/2-inch square baking pan with 1 teaspoon butter and set aside.
Melt 1 cup butter in a saucepan over low heat; stir in the sugar until thoroughly combined. Stir in the vanilla. Whisk in the eggs, one by one, whisking well between each egg. Stir in the salt and lemon zest. Transfer to a large mixing bowl.
In a separate bowl, sift the flour and baking powder together. Gently stir the flour mixture into the egg mixture. Stir in the milk, folding the batter lightly with a spatula until thoroughly combined. Pour the batter into the prepared baking pan.
Bake in the preheated oven until a toothpick inserted near the center of the cake comes out clean, 50 to 60 minutes. Let cool in pan on a wire rack for 10 minutes; invert the cake onto a second wire rack and let cool completely.