Preheat the oven to 350°F (175°C). Grease an 8×8-inch baking dish and set aside.
In a medium bowl, whisk together 1 cup all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and 3/4 cup sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the milk, 2 tablespoons lemon juice, lemon zest, and vanilla extract until combined.
Gradually add the flour mixture to the wet ingredients, mixing just until incorporated. Set the batter aside.
In a separate medium bowl, whisk together 1 1/2 cups milk, 3/4 cup sugar, eggs, 1/4 cup lemon juice, 2 tablespoons flour, and 1 tablespoon lemon zest until smooth and well combined.
Pour the lemon pudding mixture into the prepared baking dish.
Carefully spoon the lemon cake batter evenly over the pudding mixture. Do not stir.
Bake in the preheated oven for 40 to 45 minutes, or until the cake layer is golden and a toothpick inserted into the cake comes out clean.
Remove from oven and allow to cool for at least 20 minutes before serving to let the pudding set.
Serve warm or chilled as desired.