Preheat oven to 350ºF (180ºC).
Butter a 9-inch and 6-inch (or two 8-inch) cake pans with removable bottoms. Dust with flour and shake off excess.
Sift together 1 ⅓ cups all-purpose flour, ½ cup unsweetened cocoa powder, ¾ teaspoon baking soda, ½ teaspoon baking powder and ¼ teaspoon salt.
In a large bowl beat 10 tablespoons unsalted butter until creamy, about 1 minute. Add ½ cup brown sugar and ½ cup white sugar gradually and beat for 2 minutes after adding the last part.
Add 3 eggs, one at a time, beating well after each addition.
Add 1 teaspoon vanilla extract and 2 ounces unsweetened chocolate. The mixture will look curdled, that’s fine.
Add the dry ingredients in 3 parts, alternating with ½ cup buttermilk in 2 parts. Mix just until well combined.
Slowly add ½ cup boiling water and mix just until combined.
Add 4 ounces milk chocolate chips with a spatula, and divide between pans, using a bit more in the large one.
Bake for about 40 to 45 minutes, until a tester inserted in the middle of both cakes, comes out clean.
Let cool completely on a wire rack before filling and frosting. It can be made the day before and kept wrapped in plastic.