Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking pan or line it with parchment paper.
In a large mixing bowl, combine melted butter, granulated sugar, and 1 teaspoon of vanilla extract. Mix until well combined.
Add the eggs, one at a time, mixing well after each addition. Then, stir in the red food coloring.
In another bowl, whisk together the flour, cocoa powder, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour half of the brownie batter into the prepared baking pan, spreading it evenly.
In a separate bowl, beat the softened cream cheese until smooth. Add powdered sugar, lemon juice, and 1 teaspoon of vanilla extract. Mix until creamy and no lumps remain.
Spoon dollops of the cream cheese mixture over the brownie layer in the pan, then use a knife or skewer to swirl it gently into the brownie batter.
Pour the remaining brownie batter on top of the cheesecake layer, and use the knife to swirl the two layers together slightly for a marbled effect.
Bake for 30-35 minutes or until a toothpick inserted into the center comes out with a few moist crumbs (but not wet).
Allow the brownies to cool completely in the pan before cutting them into squares.
Notes
Allow brownies to cool completely for best texture before cutting.