Preheat the oven to 350°F (175°C).
Pour the melted butter into a 9-inch round cake pan and swirl to coat the bottom evenly.
Sprinkle 1/2 cup granulated sugar evenly over the melted butter in the pan.
Arrange the thinly sliced lemon slices in a single layer on top of the sugar and butter mixture, slightly overlapping them.
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 1 cup granulated sugar together until light and fluffy, about 3-4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract, fresh 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 lemon slices in the cake pan and spread gently with a spatula.
Bake for 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 upside down over the cake pan and carefully invert the cake onto the plate.
Lift the pan off gently. If some lemon slices stick to the pan, place them back on top of the cake.
Allow the cake to cool completely before slicing and serving.