Melt the butter in a saucepan over a low heat. Add the flour and stir until the mixture forms a smooth paste which leaves the sides of the pan cleanly. Cook for two to three minutes until the butter and flour make a floury ball.
Remove from the heat and gradually add the milk, stirring constantly to avoid lumps.
Bring to the boil, still stirring, and simmer for about five minutes, stirring occasionally.
When smooth and creamy, stir in the grated cheese a little at a time until melted. If the sauce is too thick, add a little extra milk.
Preheat the grill to hot.
Add the mustard, cayenne pepper and salt to the cheese sauce.
Cut the lobster horizontally down the middle. Place on a baking tray and liberally pour the cheese sauce over both halves.
Cook the lobster under the grill until the cheese is golden and bubbling.