Butter the sides and bottoms of two 9-inch round cake pans and line the bottoms with parchment paper disks cut to size.
Combine the butter and the milk in a medium saucepan and cook over low heat until the butter is completely melted. Set aside.
Sift the flour and the baking powder over a medium mixing bowl.
In the bowl of an electric mixer, whisk the eggs with the salt. Slowly add in the sugar, then the vanilla extract, while continue whisking (I like to use a stand up mixer) at medium-high speed until well mixed, about 3 minutes.
Slowly add the warm milk and butter mixture, while whisking gently. Add the flour mixture in 4 additions and continue to whisk until completely incorporated. The batter will be slightly liquid. Divide the batter evenly into the two cake pans, tap the pans gently on the counter to level their content, and place in the oven. Bake until the cakes have risen and their surface is golden, about 25-30 minutes (if you’re not sure of their doneness, insert a toothpick in the center and make sure it comes out clean).
Use a small knife to detach the cake from the sides of the pan, then invert the cakes onto cooling racks. Immediately re-invert the cakes and let cool completely, with the paper on the bottom.
Make the frosting:
In a food processor, purée 1 lb strawberries with the sugar until smooth. Transfer to a medium saucepan and bring to a boil, then reduce the heat to medium and cook, stirring frequently, until it turns darker in color and becomes syrupy, about 10 minutes. Strain through a colander, pressing on any small solids, and refrigerate for about 1 hour (or place the saucepan over an ice bath to speed up the process).
Whip the cream until soft peaks form, then reduce the speed to low and slowly stream in the strawberry puré, while continuing to whisk (you can also incorporate the strawberry puré with a rubber spatula).
Slice 1 lb strawberries into 1/4-inch thick disks.
Assemble the cake:
Remove the paper disks from the cake bottoms. With a serrated knife, cut off the tops of the cakes until leveled. Then slice each cake horizontally into two rounds. You will end up with 4 equal cake rounds.
Place one cake on a plate, cut-side up. Spread about 3/4 cup of frosting with an offset spatula and cover with a layer of sliced strawberries. Spread another 3/4 cup frosting, level with the offset spatula, and top with a second cake, cut-side down. Repeat the process with the frosting and sliced strawberries, then top with a third cake, cut-side up.
Repeat the layers of frosting and sliced strawberries and top with the last cake layer, cut-side down. Spread the remaining frosting evenly over the top and sides of the cake, using a frosting blade to evenly spread the frosting over the sides of the cake.
Top the cake with the remaining 1 lb of strawberries, cutting some of them transversally for decorative purposes. Dust lightly with icing sugar.