Preheat the oven to 350°F (175°C). Grease and flour two 8-inch (20 cm) round cake pans and line the bottoms with parchment paper.
In a medium bowl, whisk together the sifted flour, baking powder, and salt. Set aside.
In a large bowl, beat the softened butter and granulated sugar with an electric mixer 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.
Alternately add the dry ingredients and the milk to the butter mixture, beginning and ending with the dry ingredients. Mix each addition just until combined; do not overmix.
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 let them 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 egg yolks, granulated sugar, lemon juice, and lemon zest.
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. It should coat the back of a spoon.
Remove the bowl from the heat and immediately whisk in the butter pieces one at a time until fully incorporated and smooth.
Strain the lemon curd through a fine mesh sieve into a clean bowl to remove any cooked egg bits. Cover with plastic wrap pressed directly on the surface and refrigerate until chilled and thickened, at least 1 hour.
To assemble the cake, place one cake layer on a serving plate. Spread half of the chilled lemon curd evenly over the top.
Place the second cake layer on top and spread the remaining lemon curd over the top and sides of the cake.
Refrigerate the assembled cake for at least 30 minutes before serving to allow the lemon curd to set.
Slice and serve chilled or at room temperature.