Preheat oven to 350 degrees. Butter two 8-inch round cake pans, and line with parchment. Butter paper, and flour pans. Whisk together flour, baking soda, and salt in a medium bowl.
Beat together butter and sugars with a mixer until light and fluffy, about 5 minutes. Beat in yolk and eggs, 1 at a time, beating well after each addition. Beat in vanilla. Gradually add flour mixture to butter mixture in 3 additions, alternating with sour cream and milk, beginning and ending with flour. Fold in chocolate.
Divide batter between pans, and spread evenly with an offset spatula. Bake until deep golden brown and a toothpick inserted in centers comes out clean, about 40 minutes. Let cakes cool in pans 20 minutes. Invert cakes, remove pans and parchment, and let cool completely, right side up, on wire racks.
Cut each cake in half horizontally with a serrated knife. Place 1 bottom layer on a platter or cake stand, and spread evenly with 3/4 cup frosting. Repeat with a second layer and another 3/4 cup frosting. Place third layer on top, and spread with another 3/4 cup frosting. Place fourth layer on top. Spread entire cake with remaining frosting (about 2 3/4 cups), smoothing top and sides. Serve immediately, or refrigerate, uncovered, up to 2 days; if refrigerated, let cake come to room temperature before serving.
This chocolate-flecked creation with milk chocolate frosting is a great party cake. Enriched with shards of bittersweet chocolate and a luxurious frosting, it makes for a dramatic presentation and can be assembled two days ahead of time.