Preheat oven to 140 degrees celsius (fan-forced). Grease and line the base and sides of a 23cm round cake tin and set aside.
Place the white chocolate, butter, caster sugar and milk into the Thermomix bowl. Melt for 6 minutes on 50 degrees, Speed 2 (or until completely melted). Allow to cool.
Insert the butterfly, and add the eggs and vanilla extract. Mix for 10 seconds, Speed 3.
Add the plain flour and self-raising flour and mix for 10 seconds, Speed 3. Scrape down the sides of the bowl and repeat for a further 10 seconds or until combined.
Pour into the prepared tin and bake for approximately 1 hour and 45 minutes or until a skewer inserted into the middle comes out clean (see notes). Allow the cake to cool completely in the tin before removing to a serving plate.
Once the cake has completely cooled, make the buttercream by melting the white chocolate in a clean and dry Thermomix bowl for 3 minutes on 50 degrees, Speed 2. Allow the chocolate to cool to room temperature.
In a clean and dry Thermomix bowl, insert the butterfly and add the chopped room temperature butter. Beat for 1 minute on Speed 4, or until pale, smooth and creamy. While the blades are still turning on Speed 4, add the icing sugar (1 tbs at a time) and continue to mix until completely combined. Slowly add the cooled white chocolate through the MC hole and mix on Speed 1 until smooth, creamy and spreadable (see notes).
Spread the buttercream over the sides and top of the mud cake.