Heat oven to 350°F. Line an 8-inch square pan with parchment, leaving overhang.
Whisk melted butter, 0.875 cup sugar, oil, 1 tsp vanilla, food coloring, vinegar, and salt in a bowl.
Beat in the eggs one at a time until glossy and well combined.
Stir flour, cocoa, and baking powder together. Fold dry mixture into wet just until no dry streaks remain.
Spread about 2/3 of the red batter into the pan. Reserve the rest for swirling.
Beat cream cheese with 0.25 cup sugar until smooth. Mix in sour cream, 0.5 tsp vanilla, then the egg yolk.
Spoon cheesecake mixture in dollops over the red layer. Add small spoonfuls of the reserved red batter on top.
Swirl gently with a skewer or knife to create a marbled pattern. Do not overmix.
Bake 32–36 minutes, until the edges are set and the center has a slight jiggle. A few moist crumbs are fine.
Cool in the pan. Chill 30 minutes for cleaner cuts, then slice into squares and serve.