Melt butter in a saucepan over low heat and stir in flour to form a smooth paste.
Cook the mixture for three minutes until it forms a ball.
Remove from heat and gradually whisk in the milk to avoid lumps.
Boil the mixture and simmer for five minutes while stirring occasionally.
Stir in the grated cheese gradually until melted and smooth.
Preheat the grill to a high temperature.
Season the sauce with mustard, cayenne pepper, and salt.
Halve the lobster horizontally and place it on a baking tray.
Pour the cheese sauce over the lobster halves.
Grill the lobster until the topping is golden and bubbling.