Preheat the oven to 350°. Butter and flour 8 cups in a standard-size muffin pan. In a small saucepan, melt the butter. Add one-fifth of the milk chocolate and whisk until melted. Let cool slightly.
In a small bowl, whisk the cake flour with the all-purpose flour, unsweetened cocoa powder and salt. In a large bowl, using a handheld electric mixer, beat the eggs with the sugar at medium speed until pale and thick, about 5 minutes. Beat in the vanilla extract. Beat in the melted chocolate mixture and the dry ingredients in 3 additions, scraping down the bowl occasionally with a spatula. Stir in the remaining chopped chocolate.
Spoon the cake batter into the prepared muffin cups and bake for about 22 minutes, until the cakes are risen and springy. Let the cakes cool in the muffin pan for 15 minutes, then turn them out onto a wire rack and let cool for 15 minutes longer. Serve the cakes warm, with ice cream.