Beat the two eggs together in small bowl, set aside.
In large bowl, mix butter with sugar, vanilla extract, cocoa powder, salt, food coloring, and vinegar.
Whisk in the eggs.
Fold in flour, combine well but do not over-mix.
Spray 8"x8" baking pan with nonstick spray. Pour batter into pan, leaving a little extra in bowl.
Combine the cream cheese, sugar, egg yolk, and vanilla in bowl using hand mixer on medium speed.
Mix for about 1 minute until smooth.
Spoon dollops of the cheesecake filling on top of the brownie batter in the pan.
Pour extra brownie batter over top of both layers.
Using a butter knife, start at the top of the pan and go through the batter and cheesecake filling from side to side across the pan. Once you have gone from side to side all the way down, change directions and go up and down.
Bake for 28-30 minutes. They are done when a toothpick poked in the center comes out clean.