Make the cake: Preheat the oven to 350 degrees F. Butter two 8-by-2-inch-round cake pans and dust with flour, tapping out the excess. Combine the chocolate, milk and 1/2 cup water in a microwave-safe bowl. Microwave in 30-second intervals, stirring, until the chocolate is melted; whisk until smooth, then let cool. Meanwhile, sift the flour, cocoa powder, baking powder, baking soda and salt into a large bowl; set aside.
Beat the butter in a large bowl with a mixer on medium speed until creamy, about 4 minutes. Add the vegetable oil, brown sugar, granulated sugar and vanilla; beat until fluffy, about 4 more minutes. Add the eggs, one at a time, beating until incorporated. Reduce the mixer speed to low and add the flour mixture in three batches, alternating with the melted chocolate mixture; beat until just incorporated.
Divide the batter between the prepared pans and smooth the tops. Bake until a toothpick inserted into the centers comes out clean, 45 to 50 minutes. Let cool 20 minutes in the pans, then run a knife around the edges and invert onto a rack to cool completely.
Reheat the oven to 350 degrees F. Line a rimmed baking sheet with foil. Using a long serrated knife, cut each cake in half crosswise to make 4 layers. Crumble one of the layers into small pieces on the baking sheet. Bake until dry, about 20 minutes; let cool completely. Cover with parchment paper and crush with the bottom of a saucepan to make coarse crumbs; set aside.
Make the frosting: Combine the chocolate, corn syrup and 1/2 cup hot water in a large heatproof bowl set over a pot of simmering water; stir until melted, about 5 minutes. Remove the bowl from the pot and let cool 10 minutes. Transfer to a stand mixer fitted with the whisk attachment and start beating on high speed; beat in the vanilla, then the butter, 1 tablespoon at a time, until fluffy. Gradually add the confectioners' sugar and beat until creamy. Refrigerate, stirring occasionally, until thick enough to spread, about 10 minutes. (If the frosting is too thick, microwave in 10-second intervals for up to 30 seconds to soften.)
Put one cake layer on a platter; spread 1 heaping cup frosting on top. Add another cake layer, more frosting and the last cake layer. Cover the top and sides with the remaining frosting. Press the reserved cake crumbs all over the cake.