Red Velvet Bread Pudding

For the cake:

2 cups sugar
1/2 pound (2 sticks) butter, at room temperature
2 eggs
2 tablespoons cocoa powder
1 ounce red food coloring
2 1/2 cups cake flour
1 teaspoon salt
1 cup buttermilk
1 teaspoon vanilla extract
1/2 teaspoon baking soda
1 tablespoon vinegar

For the bread pudding mix:

3 cups half-and-half
3 large eggs
1 large egg yolk
pinch of salt
1 teaspoon vanilla extract
8 ounces cream cheese, at room temperature
3/4 cup powdered sugar, plus more for sprinkling