Melt 10g of butter in a frying pan and add the spring onions. Sprinkle with half the lemon juice, powdered sugar, some water, salt and pepper and simmer almost covered until the spring onions are tender and the juice is almost completely absorbed.
Roll up each fillet of sole on itself and place them in an oiled ovenproof dish, one against the other.
Bake for 10 minutes.
Peel and finely chop the shallot. Brown it in a pan with 10g of butter, once translucent, add the Champagne and the rest of the lemon juice. Allow to reduce almost to dryness. Add the cream and simmer for a few moments.
Remove from the heat, add the remaining butter in small cubes and mix to obtain a foamy butter.
Remove the sole from the oven, spread some of the foaming butter on the bottom of the plates and add the sole rolls on top.
Add the caviar on the sole and the foamy butter, sprinkle with nasturtium leaves and serve immediately.