Put the butter and chocolate in a medium-sized heatproof bowl. Melt the ingredients either using the bain-marie method (suspending the bowl over a pan of barely simmering water) or heat on moderate power in a microwave, stirring every 30 seconds, until melted. Beat until smooth.
Add the remaining caster sugar and the dark muscovado sugar to the bowl. Use a balloon whisk to mix them in then beat in the eggs and egg yolks along with the vanilla extract.
Sieve the cocoa powder and flour into the bowl and add the salt. Use a balloon whisk to mix the ingredients in until just combined.
Remove ¼ of the brownie batter, place in a small bowl and set aside.
If including white chocolate chips stir them into the remaining brownie now then ladle the batter into a lined baking tin and smooth out.
Top with the matcha cheesecake mixture, spreading it out to cover the base layer and then dollop the reserved brownie batter over the top - do not spread this out.
Use a cocktail pick to gently swirl the ingredients together
Bake for around 30 minutes, testing for doneness 5 minutes before the end of the cooking time. Poke a cocktail pick into the centre of the bake. If liquid batter is evident, keep on baking for a few minutes more and test again. Of not, it is cooked sufficiently (a few crumbs may attach to the pick, this is fine).
Let the matcha brownies cool in the tin completely before unmolding and cutting into portions.