Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, beat the softened butter and granulated sugar until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the lemon zest and vanilla extract.
Alternately add the dry flour mixture and the milk mixed with lemon juice to the butter mixture, beginning and ending with the flour mixture. Mix until just combined.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 30 to 35 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 10 minutes, then transfer to a wire rack to cool completely.
To prepare the lemon glaze, whisk together the powdered sugar, fresh lemon juice, and lemon zest until smooth.
Once the cake is completely cooled, drizzle the lemon glaze evenly over the top.
Allow the glaze to set for 15 minutes before slicing and serving.