Place flour and milk into the Thermomix bowl and blend 5 sec/speed 6.
Cook 15 min/80°C/speed 4. Pour into bowl, cover with plastic wrap, pushing plastic down onto surface of mixture. Cool completely.
In clean and dry Thermomix bowl, add sugar and mill 15 sec/speed 9. Add butter and Chop 20 sec/speed 7.
Aerate butter and sugar 1-2min/Butterfly/speed 3/MC off. Scrape down sides of bowl as necessary.
Add vanilla, colour and peppermint oil and beat 6 sec/Butterfly/speed 4. Add cooked and cooled roux mixture and continue to beat for around 2-3 minutes/Butterfly/speed 3 until very smooth and white. Cover and refrigerate for 15 minutes.
To assemble the 3 layers of mud cake, decide which colour goes on the base. Spread a little frosting onto the cake stand you will use and place the first layer on to it. Press down.
Pipe enough of the frosting on the first layer and spread to create 1 cm thick layer to the edges. Sprinkle with a few of the chopped candy canes. Set it in the fridge for 10 minutes. Top with the second layer and repeat. Place the final layer on top and use the remaining frosting to cover the top and sides of the cake.
Decorate the bottom of the cake with crushed candy canes and garnish as you wish on top of the cake.