Preheat oven to 350°F (180°C). Spray a 13×9-inch baking pan with baking spray with flour. Line pan with parchment paper, letting excess extend over all sides of pan.
In the bowl of a stand mixer fitted with the paddle attachment, beat 1 cup (227 grams) butter, granulated sugar, and oil at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition. Beat in vanilla.
In a medium bowl, whisk together flour, baking powder, and salt. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition.
In a large microwave-safe bowl, heat chocolate on high in 30-second intervals, stirring between each, until completely melted. Stir in cocoa and remaining ¼ cup (57 grams) butter until butter is completely melted and mixture is well combined. Stir in one-third of vanilla batter (about 2 2/3 cups or 620 grams) until combined.
Spread half of remaining vanilla batter (about 2 2/3 cups or 620 grams) in bottom of prepared pan. Scoop chocolate batter and remaining vanilla batter by heaping tablespoonfuls (about 20 grams each) in a checkerboard pattern into pan (5 scoops wide along short side). Tap pan on a kitchen towel-lined counter 2 to 3 times to help level batter and fill pan. Using a knife, swirl batters in a zigzag pattern lengthwise and widthwise. Tap pan to help level batter and fill in any gaps.
Bake until a wooden pick inserted in center comes out with a few moist crumbs and an instant-read thermometer inserted in center registers 200°F (93°C) to 205°F (96°C), 50 to 55 minutes, covering with foil to prevent excess browning, if necessary. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool completely on a wire rack.
Transfer cooled cake to a serving platter. Spread Buttermilk Chocolate Frosting on top of cake. Using a small offset spatula, texture and swirl as desired. Refrigerate, covered, for up to 5 days.