Preheat oven to 400F.
Bring a pan of salted water to a boil and cook the carrots until tender.
To make the Bechamel sauce melt the butter in a small pan, then stir in the flour and cook over a low heat for 1-2 minutes, stirring continuously to make a thick paste. Gradually add the milk, stirring well all the time, then increase the heat and continue to stir until the sauce thickens. Add salt, pepper and nutmeg to taste.
Drain the carrots and mash them finely or puree in a food processor.
Whisk the egg whites until they form very soft peaks.
Stir the sauce and egg whites into the carrots and mix well. Pour into an oven-proof dish and bake for 10 minutes. Dot the top with a little more butter, scatter the cheese around the top and then bake for a further 10 minutes. The top will form a crust and the carrot will be light and delicious.