Step 1

For the cake: Position an oven rack in the middle of the oven and preheat to 350 degrees F. Spray a 9-by-13-inch baking pan with cooking spray and line the bottom with parchment.

Step 2

Place the chocolate in a medium microwave-safe bowl and microwave in 20-second intervals, stirring in between, until the chocolate is fully melted. Set aside.

Step 3

Whisk together the flour, baking powder, baking soda and salt in a medium bowl and set aside.

Step 4

Beat the granulated sugar, butter and oil in a large bowl with an electric mixer on medium-high speed until fluffy and pale, about 4 minutes. Reduce the mixer speed to low and add the vanilla and then the eggs one at a time, scraping down the sides of the bowl with a rubber spatula and mixing well after each addition. Add one-third of the flour mixture and beat on low speed until just combined. Add half of the milk and beat until combined. Repeat alternating additions of the flour mixture and milk, scraping down the sides of the bowl after each and beating until the last addition of flour is fully incorporated.

Step 5

Transfer one-quarter of the batter to the bowl with the melted chocolate and stir until the chocolate is evenly incorporated.

Step 6

Pour half of the remaining vanilla batter into the prepared cake pan and smooth out to the edges with an offset spatula. Scoop large spoonfuls of the chocolate batter on top of the vanilla batter and then pour the remaining vanilla batter on top. Using a skewer or butter knife, carefully swirl the chocolate batter into the vanilla batter to create a marbled effect.

Step 7

Bake, rotating the pan halfway through, until a toothpick inserted into the center of the cake comes out clean, about 35 minutes. The top of the cake should be springy to the touch and the vanilla spots should be lightly golden. Cool the cake in the pan for 10 minutes, then use a paring knife to loosen the sides of the cake from the pan and invert onto a wire rack to cool completely before frosting.

Step 8

For the frosting: Beat the butter in a medium bowl with an electric mixer on medium-high speed until smooth and creamy, about 3 minutes. Add the cocoa powder, cream, vanilla, salt and 1 1/2 cups of the confectioners' sugar and mix on low speed until fully incorporated. With the mixer still on low, gradually add the remaining confectioners' sugar and mix well. Increase the mixer speed to medium high and beat until the frosting is light and fluffy, about 2 minutes; the frosting will lighten in color as air is incorporated. (See Cook's Note.)

Step 9

Using an offset spatula, spread the frosting in a thick, swooping layer all over the top of the cooled cake.