For the sea bass, preheat the oven to 200C/180C Fan/Gas 6.
Take a piece of cardboard and trace the rough shape of your sea bass onto it, adding 2.5cm/1in border. Cut the fish shape from the cardboard, then wrap it in 2 layers of aluminium foil. Place the foil-wrapped cardboard onto a baking tray and place the sea bass on top.
Whisk the egg whites in a bowl until soft peaks form when the whisk is removed. Add the salt and continue to whisk until the egg whites are firm and smooth, and stiff peaks form when the whisk is removed.
Using a palette knife, spread the egg-white mixture evenly over the whole fish all the way down to the foil to make an airtight seal. If desired, smooth the surface of the egg whites, then lightly draw the shape of the head, mouth and eye of the fish with the edge of the palette knife, and use the end of the palette knife to gently make indentations in the egg-white mixture that resemble scales.
Bake in the oven for 20 minutes. To check whether the fish is cooked through, insert a cake tester or thin skewer into the thickest part of the fish for 10 seconds. It should feel hot when you take it out. If it is only lukewarm, cook the fish for a little longer and test again.
Meanwhile, for the butternut squash purée, place half of the chopped shallot into the hole in each butternut squash half. Grate in a little garlic, drizzle over the oil, then season, to taste, with salt and freshly ground black pepper.
Place the butternut squash onto a baking tray and roast in the oven with the sea bass for 25-30 minutes, or until the squash is tender.
Remove the squash from the oven and scoop the flesh into a food processor. Blend until completely smooth, then season again if necessary. Set aside until needed.
For the Champagne sauce, heat 50g/2oz of the butter in a saucepan over a medium heat. When the butter is foaming, add the vegetables and fry for 4-5 minutes, or until just softened but not coloured.
Pour in half of the Champagne and bring to the boil. Reduce the heat and simmer until the mixture resembles syrup.
Pour in the fish stock, and simmer for 4-5 minutes, until the volume of liquid has reduced by half.
Pour in the cream and simmer for 4-5 minutes, until the volume of liquid has reduced by half.
Strain the sauce through a fine sieve into a clean saucepan. Season with salt and pepper. Place the pan over a low heat and whisk in the remaining butter and the lemon juice. Keep warm until ready to serve, then stir in the remaining Champagne.
Just before serving, bring the salt-crusted sea bass to the table and crack the crust using a sharp knife. Carefully peel away the salt crust in as few pieces as possible, discarding any salt crumbs remaining on the fish. Carefully remove the fillets from the fish carcass, discarding any large bones as you go.
To serve, spread a circle of the butternut squash purée into the centre of four serving plates. Place some of the filleted sea bass on top. Spoon over the Champagne sauce.