Preheat oven to 325°F. Line an 8×8-inch metal baking pan with parchment paper, leaving a 2-inch overhang at all sides.
Place eggs and granulated sugar in the bowl of a stand mixer fitted with the whisk attachment and beat on medium-high speed for 2-3 minutes, until creamy and lemon-coloured, or use an electric hand mixer.
Meanwhile, heat butter in a saucepan until completely melted and very hot (it will start crackling). Remove from heat and whisk in icing sugar, cocoa powder and salt until smooth.
When the eggs are pale and thick, add the warm cocoa mixture and beat for 30 seconds until smooth and shiny. Add the flour and mix it in on low speed. It should be smooth and glossy, like thick hot fudge. Fold in chocolate chips.
Pour the batter into your prepared pan bake for about 25 minutes, until a toothpick inserted in the center comes out with moist crumbs attached. Make sure you do not over-bake these or they will be dry. Transfer pan to a wire rack and let brownies cool completely in the pan.
To make the white chocolate ganache, combine the chopped white chocolate, cream and peppermint extract in a heatproof bowl over a saucepan with an inch of barely simmering water and stir until completely melted and smooth. Pour it over the brownie in the pan and place in the fridge for at least 30 minutes until set.
To make the dark chocolate ganache, heat cream until it just comes to a simmer over the stove or in the microwave. Immediately pour it over the chopped chocolate, cover the bowl and let stand for 2 minutes. Stir until silky smooth and glossy. If any chocolate remains unmelted, just heat very gently in the microwave for a few seconds and keep stirring. DO NOT OVER HEAT. Pour it over the peppermint ganache layer and refrigerate for at least 1 hour before lifting out and cutting into squares.