Step by step recipe:

1. 20 min.

   Boiling the langoustines

   Boil the langoustiens for 30 seconds in boiling salted water, then cool under running cold water.
   
   Shell them and set aside the tails. Keep half the heads for making the fumet.
2. 5 min.

   The langoustine fumet

   Prepare 1 Bouquet garni and finely chop 30 g onion.
3. 2 min.
   Melt 30 g butter in a large saucepan on medium heat, then add the chopped onion, salt and pepper.
   
   Cook for 1 minute without colouring.
4. 1 min.
   Add the langoustine heads...
5. 2 min.
   ... and crush briefly with the end of a rolling pin, for example.
6. 1 min.
   Pour in 200 ml water and 200 ml dry white wine, then add the bouquet garni.
7. 5 min.
   Bring to the boil, then turn down the heat to its lowest setting and leave to reduce by half.
   
   This simmering will create the "fumet", as the liquids (water and wine) mingle with the flavours of the bouquet garni and langoustine heads.
8. 40 min.

   The vegetable brunoise

   While the fumet is reducing, rinse and peel 2 carrots and 3 turnips, then dice very small ("fine brunoise").
   
   Rinse the courgette and cut the part with the green skin into tiny dice (you can use the leftover courgette in another recipe).
   
   I have to admit, this is a long, arduous task!
9. 5 min.
   Bring a large saucepan of salted water to the boil.
   
   Immerse a fine metal strainer, which will enable you to remove the vegetables quickly.
10. 1 min.
    Toss the diced courgettes into the water, wait until the pan comes back to the boil, then remove them immediately by simply lifting out the strainer.
11. 1 min.
    Cool the diced courgettes under running cold water to stop the cooking...
12. 2 min.
    ...drain thoroughly, then tip out onto absorbant paper to wait.
13. 5 min.
    Do the same with the carrots and turnips, which can be cooked together, and for 30 seconds longer than the courgettes.
    
    Note: Don't throw this cooking water away - it is now worth keeping.
14. 4 min.

    Second-stage cooking of langoustines

    Melt 30 g butter in a frying pan on high heat. When good and hot, add the langoustine tails.
    
    Fry for a minute or two until they start to brown, then set aside.
15. 3 min.

    The gratin cream sauce

    Let's go back to the fumet, which by now should be reduced by half. Strain into a bowl and discard everything but the liquid.
16. 2 min.
    Pour this back into the pan (after rinsing out), but be careful to keep back the dregs of the fumet which will be full of sand and should be discarded.
17. 8 min.
    Put the fumet back on medium heat and leave to reduce by half again.
18. 3 min.
    Add 250 ml liquid cream, salt lightly, pepper and then check the seasoning.
19. 2 min.
    Bring to the boil, take off the heat and and 20 g cornflour (mixed with a little cold water) all at once while beating with a whisk.
    
    The mixture should thicken rapidly. This will give us enough sauce to cover the gratin. Set aside.
20. 4 min.

    Assembling the gratin

    Preheat the oven to 390°F (200°C).
    
    Butter a gratin dish and spread the mixed diced vegetables in an even layer.
21. 2 min.
    Arrange the sautéed langoustine tails on top.
22. 2 min.
    Pour the sauce over.
23. 15 min.

    Cooking and serving

    Bake for about 15 minutes, until the top is nicely browned (finish under the grill if necessary).
24. 
    Best served plated up (on hot plates): try to keep the langoustines on top and the vegetables underneath.