Preheat the Oven: Set your oven to 180°C (160°C fan) and line a 9x9" square tin with parchment paper.
Make the Brownie Batter:
Beat together the butter and sugar until well combined.
Add the vanilla extract and red food coloring. It's easier to mix these two together first into a thinner paste before adding them to the butter and sugar mixture.
Add the eggs one at a time, beating after each addition.
Mix in the cocoa powder and plain flour, then beat until fully incorporated.
Prepare the Cheesecake Swirl:
In a separate bowl, beat the cream cheese, sugar, and vanilla extract until smooth.
Add the egg yolk and mix again until well combined.
Assemble the Brownies:
Pour about 90% of the brownie mixture into the prepared tin.
Dollop the cream cheese mixture over the brownie batter, then add the remaining brownie mixture on top.
Use a spatula or knife to swirl the mixtures together.
Bake:
Bake in the preheated oven for 28-35 minutes, or until a skewer inserted into the center comes out clean. (Mine baked for 30 minutes.)
Cool:
Leave the brownies to cool in the tin for 20-30 minutes, then transfer to a wire rack to cool completely.