Brownies: Melt the butter in the microwave, then whisk in the vanilla extract, sugar, and Bailey's. Add the eggs, one at a time, whisking between each addition. Add dry ingredients: flour, cocoa powder, salt, and espresso powder, stir until just combined. Spread evenly across the bottom of the prepared baking dish.
Cheesecake: Add the room temperature cream cheese to a mixing bowl or the bowl of a stand mixer. Using a hand mixer, cream the cream cheese until smooth. Add the sugar and Bailey's, continuing to beat until light and fluffy. Add in the eggs, one at a time, beating between each addition and scraping the sides of the bowl as needed, until fully combined. Spread the cheesecake batter on top of the brownie batter evenly.
Bake for 38-42 minutes until mostly set (the center may jiggle a little still, that's ok!). Once you've removed the bars from the oven, let them cool for 30-45 minutes at room temperature. Then, place in the fridge to chill for approximately 2 hours.
When ready to serve, combine all the ganache ingredients in a microwave-safe bowl (you can also use a double boiler). Microwave the ingredients in at least 2, 15-second increments, stirring between each time. At this point, it should come together smoothly after several good stirs; if not, microwave for an additional 10-15 seconds. Don't cook too long! Drizzle or spread the ganache on top of the bars, cut as desired, and enjoy!
You may have some ganache leftover if drizzling across brownies (opposed to spreading in a layer).The vegetable oil smooths out the ganache and adds a shine, but feel free to omit if desired.