01 - Preheat oven to 350°F. Grease and line a 9x5-inch loaf pan with parchment paper.
02 - Whisk together flour, baking powder, baking soda, and salt in a medium bowl; set aside.
03 - Beat softened butter and sugar until light and fluffy, approximately 3 minutes.
04 - Incorporate eggs one at a time, beating well after each addition, then stir in lemon zest and vanilla extract.
05 - Stir lemon juice into the batter; a slightly curdled appearance is normal.
06 - Add half of the dry ingredients to the wet mixture, mixing until combined. Pour in milk and mix briefly, then fold in remaining dry ingredients. Avoid overmixing.
07 - Pour batter into prepared pan and smooth the top evenly.
08 - Bake for 45 to 55 minutes or until a toothpick inserted in the center comes out clean.
09 - Allow to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
10 - Whisk powdered sugar with 2 tablespoons lemon juice; add more juice if a thinner consistency is desired. Drizzle over cooled loaf before slicing.