Preheat the oven to 350°F (175°C). Grease a 9-inch round baking dish or ovenproof dish.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, beat 1/2 cup granulated sugar with the egg yolks until pale and thick.
Add the melted butter, whole milk, lemon juice, lemon zest, and vanilla extract to the egg yolk mixture and mix until combined.
Gradually fold the dry ingredients into the wet ingredients until just combined.
In a separate clean bowl, beat the egg whites until soft peaks form. Gradually add 1/2 cup granulated sugar and continue beating until stiff peaks form.
Gently fold the beaten egg whites into the batter, preserving as much air as possible.
Pour the batter into the prepared baking dish and spread evenly.
Bake for 25-30 minutes or until the top is golden and a toothpick inserted in the center comes out clean.
While the cake is baking, prepare the lemon pudding base: In a medium saucepan, whisk together 1/2 cup granulated sugar, cornstarch, and a pinch of salt.
In a separate bowl, whisk the egg yolks with 2 cups whole milk, lemon juice, and lemon zest until smooth.
Slowly pour the egg yolk mixture into the saucepan with the sugar and cornstarch, whisking constantly.
Cook over medium heat, stirring constantly, until the mixture thickens and comes to a gentle boil, about 5-7 minutes.
Remove the pudding from heat and let it cool slightly.
Once the cake is done, remove it from the oven and allow it to cool for 10 minutes.
Pour the warm lemon pudding evenly over the top of the cake, allowing it to soak in.
Refrigerate the assembled lemon cake lemon pudding for at least 2 hours before serving to allow flavors to meld.
Serve chilled or at room temperature.