1. Pour the champagne into a saucepan with the shallots and reduce by 3/4, season with salt and pepper.
2. Strain the champagne to remove the shallots and keep the juice warm.
3. Wash and peel all the vegetables.
4. Slice the carrots, sweet potatoes, and potatoes into 5mm thick slices.
5. Cut the leeks into rounds.
6. In a large pot, heat a large amount of water with the vegetable stock cubes.
7. Put the carrots in boiling water for 2 minutes, then add the potatoes and leeks, followed by the sweet potatoes 1 minute later.
8. Continue cooking for 2 to 3 minutes, then remove the vegetables with a slotted spoon and keep the pot on the heat.
9. Boil the champagne juice, then add the cornstarch, previously mixed with a little water, until the sauce has a coating consistency.
10. Add the butter while whisking and keep warm. Whisk regularly. Taste and adjust the seasoning if necessary.
11. Place the lobsters in boiling water for 3 to 5 minutes, remove them from the water, then cut them in half.
12. Remove the meat from the tail, slice it, and place it back in the shell. Place a slice of vegetable between each slice of meat.
13. Arrange on plates, distribute the remaining garnish, and add the sauce.
14. Serve immediately.