Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20 cm) round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, 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 and vanilla extract until combined.
In a small bowl, combine the whole milk and fresh lemon juice. Let it sit for 2 minutes to slightly curdle, creating a buttermilk substitute.
Alternately add the dry flour mixture and the milk-lemon mixture to the butter mixture, beginning and ending with the dry ingredients. Mix just until combined after each addition; do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 30-35 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.
In a small bowl, whisk together the powdered sugar and 1 tablespoon fresh lemon juice until smooth to make the glaze.
Once the cake is completely cooled, drizzle the lemon glaze evenly over the top.
Slice and serve.