Preheat the oven to 180C/350F/Gas 4.
In a bowl, cream the butter and brown sugar together until pale and fluffy. Gradually add the eggs and then fold in the flour, carrot and cinnamon. Finally stir in the milk.
Butter a 25cm/10in cake tin, pour in the cake mixture and smooth the surface. Place into the oven and bake for 15-20 minutes, or until springy to touch on the top and a skewer comes out clean when inserted into the middle of the cake. Remove from the oven and leave to cool. Loosen the edges of the cake by running a sharp knife around the inside edge of the tin, then turn out onto a serving plate.
Place the icing sugar into a small bowl. Add a little water, a teaspoon at a time, and stir to make a thin icing. Drizzle the icing over the cake, cut into slices and serve.
