Preheat your oven to 325°F (160°C). Grease two 8-inch (20 cm) round cake pans and line the bottoms with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt.
In a large mixing bowl, beat 1 cup (225 g) of softened butter and the granulated sugar until light and fluffy. Gradually mix in the beaten eggs and 2 teaspoons of vanilla extract.
Add half of the flour mixture to the butter mixture, followed by the milk, and then the remaining flour. Stir until just combined.
Fold the finely chopped walnuts into the batter, ensuring they are evenly distributed.
Divide the batter evenly between the prepared cake pans. Bake for 30 to 35 minutes, or until a wooden skewer inserted into the center comes out clean.
Let the cakes rest in the pans for 2 minutes, then invert them onto a wire cooling rack to cool entirely.
For the frosting, stir 2/3 cup (150 g) of softened butter, powdered sugar, and 1 teaspoon of vanilla extract together with a spoon. Once combined, use an electric mixer to beat the frosting until fluffy and pale.
Place one cooled cake layer on a plate, spread a layer of frosting on top, then add the second cake layer. Frost the top and sides of the cake as desired.