THE CUPCAKES: Preheat oven to 180°C and place cupcake papers in 18 holes of two 12-hole 1/3 cup capacity cupcake tins.
Peel beetroot and shop into large pieces. Puree in a blender or food processor until you have a smooth paste. Set aside until needed.
Place chocolate into a small microwave-safe bowl and microwave in 20-second bursts until melted. Stir between each burst. Once melted, dissolve instant coffee in chocolate and set aside to cool.
In a large bowl whisk together buckwheat flour, brown rice flour, almond meal, cocoa powder, salt, baking soda, and baking powder.
In a medium bowl whisk together eggs, caster sugar, brown sugar, vanilla, milk, and vegetable oil until smooth.
Slowly whisk in pureed beetroot and melted chocolate until smooth and fully incorporated. Switch to a wooden spoon and pour wet ingredients into dry ingredients. Mix until just combined. Spoon into prepared cases filling them 2/3 of the way.
Place trays in oven and bake for 20 to 22 minutes or until the tops of the cupcakes bounce back when lightly touched. Allow to cool completely before frosting.
THE FROSTING: place butter in a large bowl and using a handmixer beat until pale and fluffy. Add remaining ingredients and beat until frosting is fluffy and aerated, about 3 to 4 minutes. Spoon frosting into a piping bag fitted with your choice of tip and decorate cooled cupcakes as desired. You could also use a knife to frost cupcakes rather than a piping bag. Store in an airtight container in the refrigerator for up to 3 days.