Preheat the oven to 350°F (175°C). Grease and flour two 8-inch (20 cm) round cake pans.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In a large bowl, using an electric mixer, beat the softened butter and granulated sugar on medium speed 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.
With the mixer on low speed, alternately add the dry ingredients and milk to the butter mixture, beginning and ending with the dry ingredients. Mix just until combined.
Divide the batter evenly between the prepared cake pans and smooth the tops with a spatula.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cakes from the oven and allow them to cool in the pans for 10 minutes. Then, turn them out onto wire racks to cool completely.
While the cakes are baking, prepare the lemon curd: In a medium heatproof bowl, whisk together the 4 eggs, granulated sugar, lemon juice, and lemon zest until smooth.
Place the bowl over a saucepan of simmering water (double boiler), making sure the bottom of the bowl does not touch the water.
Cook the mixture, whisking constantly, until it thickens and reaches about 170°F (77°C), about 8-10 minutes. The curd should coat the back of a spoon.
Remove the bowl from heat and immediately whisk in the butter pieces until fully melted and incorporated.
Strain the lemon curd through a fine-mesh sieve into a clean bowl to remove zest and any cooked egg bits.
Cover the lemon curd with plastic wrap pressed directly onto the surface and refrigerate until chilled, at least 1 hour.
To assemble the cake, place one cake layer on a serving plate. Spread a generous layer of lemon curd over the top.
Place the second cake layer on top and spread lemon curd over the top and sides as desired.
Refrigerate the assembled cake for at least 30 minutes before serving to allow the lemon curd to set.