Preheat the oven to 350°F (175°C).
Pour the melted butter into a 9-inch round cake pan, spreading it evenly.
Sprinkle 1/2 cup granulated sugar evenly over the melted butter in the pan.
Arrange the lemon slices in a single layer over the sugar, slightly overlapping if needed.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and 1 cup granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract, lemon juice, and lemon zest until combined.
Alternately add the flour mixture and milk to the butter mixture, beginning and ending with the flour mixture. Mix until just combined.
Pour the batter evenly over the arranged lemon slices in the cake pan.
Bake for 35 to 40 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.
Run a knife around the edges of the pan to loosen the cake.
Place a serving plate over the cake pan and carefully invert the cake onto the plate.
Remove the cake pan and allow the cake to cool completely before serving.