1. Preheat your oven to 350°F (175°C) and line a 9×13-inch baking pan with parchment paper.
2. In a medium saucepan, melt the butter over low heat. Remove from heat and stir in the granulated sugar until smooth.
3. Add 3 eggs, one at a time, beating well after each addition. Stir in 2 teaspoons of vanilla extract.
4. In a separate bowl, whisk together the flour, cocoa powder, and salt. Gradually fold the dry ingredients into the wet mixture until just combined.
5. Pour the brownie batter into the prepared pan, spreading it evenly with a spatula.
6. In a large bowl, beat the cream cheese until creamy. Add the remaining egg, sour cream, powdered sugar, lemon juice, and the last teaspoon of vanilla extract. Beat until smooth and well combined.
7. Pour the cheesecake mixture over the brownie layer, smoothing out the top.
8. Bake in the preheated oven for 35-40 minutes, or until the cheesecake is set and the edges are lightly golden.
9. Allow the bars to cool completely before slicing — patience is key here!

Pro tip from my grandmother: If you find yourself impatiently waiting for these to cool, a quick stint in the fridge can speed up the process. But remember, good things come to those who wait.