Preheat your oven to 350 degrees F (180 degrees C).
Put the finely grated carrot into a fine mesh strainer. Hold the strainer over a sink and squeeze the grated carrot to remove as much liquid as possible. Place into a medium bowl.
To the bowl, add the melted oil, egg and yogurt/buttermilk and stir in. Add the next 6 ingredients (flour through to cocoa powder) and stir in. Lastly, stir in the hot water until just combined.
Divide the batter between 12 lined muffin cups. Bake for 20 minutes (or until a toothpick comes out clean). Cool on a wire rack.
For the frosting, combine all the ingredients in a small bowl until smooth. Use about 2 tsp to frost each cupcake.