Preheat the oven to 350°F (175°C). Grease and flour a 9×5 inch loaf pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and granulated sugar together with an electric mixer on medium speed until light and fluffy, about 3-4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest, lemon juice, and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients in three additions, alternating with the milk, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake for 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 15 minutes. Then transfer the cake to a wire rack to cool completely.
To prepare the lemon glaze, whisk together the powdered sugar and 2 tablespoons fresh lemon juice in a small bowl until smooth.
Once the cake is completely cooled, drizzle the lemon glaze evenly over the top of the loaf.
Sprinkle the 1 teaspoon lemon zest over the glaze for decoration.
Allow the glaze to set for 10-15 minutes before slicing and serving.