Preheat the oven to 350 degrees F. Grease with butter or cooking spray a 13×18 jelly roll pan and set aside.
In a large bowl or the bowl of your stand mixer, add the flour, sugar, cocoa powder, salt, and baking powder. Stir to combine.
In a small bowl add the eggs, oil, 1/4 cup milk, and vanilla. Whisk the wet ingredient to combine.
Make a well in the center of the dry ingredient and pour the wet ingredients into the center. Mix until well combined or with the cookie paddle attached of a stand mixer, mix on medium-low speed until well combined. The batter will be very thick.
Remove about 3/4 cup of the batter and put it in a small bowl (or the bowl the wet ingredients just came out of) and set aside. Spread the rest of the batter in the bottom of the greased jelly roll pan.
To make the cheesecake layer, combine the cream cheese and sugar in the bowl of your stand mixer and beat on medium speed until well combined and fluffy, about 3 minutes.
Add the eggs and vanilla and beat another minute until well combined. Spread the cheesecake layer evenly on top of the brownie layer in the pan.
Add the remaining 2 tablespoons of milk to the batter that you saved in the small bowl. Use a fork to whisk and combine it well. Drop the now thinned batter by small spoonfuls over the cheesecake layer and then use a toothpick to marble the cheesecake and thinned batter together. You just pull the toothpick across the top of the dessert with a bit of swirling motion. Do this until it is marbled as much as you like.
Bake the bars for 28-30 minutes, rotating the pan halfway through baking.
Let the bars cool in the pan for 20 minutes before cutting.
Notes
Dark cocoa powder, such as Hersey's Special Dark, is amazingly chocolaty without being bitter. It's a great thing to have on hand and I now use it in 90% of recipes that call for cocoa.
Ensure cream cheese and eggs are at room temperature for a smoother cheesecake layer and easier mixing.
Allow the brownies to cool in the pan for 20 minutes before cutting into squares. This helps them set and makes for cleaner cuts when serving.
Grease the jelly roll pan generously to prevent sticking and ensure easy removal of the brownies after baking.
Use a toothpick or knife to gently swirl the brownie and cheesecake layers together. Avoid over-mixing to maintain distinct layers.