Prepare all the ingredients.

Thinly slice the shallots.

Sweat them in a knob of butter for a few seconds.

Add white wine...

...tarragon sprig...

...fish stock...

... and reduce by half.

Add a good spoonful of mustard to the bechamel sauce...

...and mix thoroughly.

Place the lobsters on a cutting board and split them in half beginning with the head...

...then the tail. It is easier to cut the tail when folded backward.

Separate the 2 parts.

Remove the guts.

Place the lobsters on a baking sheet. Add salt, pepper and paprika. Add a few knobs of butter.

Bake in warm oven and frequently sprinkle with butter (around 12 mn).

Incorporate the reduced preparation, the bechamel sauce with mustard.

Mix well using a whisk.

Strain using a fine sieve or a chinois.

When finished...

...remove the meat and the roe from the shells.

Cut them into pieces.

You may add some fried button mushrooms (optional).

Place a layer of sauce into the lobster tail , fried mushrooms...

...pieces of roe and meat...

...and coat with the sauce.

Sprinkle with grated emmental...

...a few knobs of butter...

...and leave to broil in the oven or in the broiler to obtain a regular gratinée.

Serve immediately.