Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking dish.
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 3/4 cup granulated sugar with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the milk, lemon zest, lemon juice, and vanilla extract until combined.
Gradually add the flour mixture to the wet ingredients, mixing on low speed until just combined.
Pour the batter into the prepared baking dish and spread evenly.
Bake for 25 to 30 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, prepare the lemon pudding topping. In a medium saucepan, whisk together 1 cup granulated sugar, cornstarch, and salt.
Gradually whisk in the 1 1/2 cups milk until smooth.
Cook over medium heat, whisking constantly, until the mixture thickens and comes to a boil, about 5 to 7 minutes.
In a small bowl, lightly beat the egg yolks. Gradually whisk about 1/2 cup of the hot milk mixture into the yolks to temper them.
Return the tempered yolks to the saucepan and cook for 2 more minutes, stirring constantly.
Remove the saucepan from heat and stir in the lemon juice and butter until smooth.
Once the cake is done baking, remove it from the oven and immediately pour the hot lemon pudding evenly over the cake.
Return the dish to the oven and bake for an additional 10 minutes.
Remove from the oven and allow to cool completely at room temperature before serving.