Preheat the oven to 190°C.
Mix the egg whites and salt in a bowl to form a paste.
Coat each large beetroot in a 2cm thick layer of salt paste.
Bake the salt-crusted beetroot on a tray for 45 to 50 minutes.
Place baby beetroot in a tray with 2cm of water, cover with foil, and bake.
Remove all beetroot from the oven and let them cool.
Discard the salt crust, peel the beetroot, then slice half and chop half.
Peel and quarter the baby beetroot.
Heat a deep fat fryer to 190°C.
Boil 125ml of water and butter in a saucepan for one minute.
Remove from heat, add flour, and beat until the mixture is smooth.
Cook the dough over heat for two minutes.
Remove from heat and beat in the eggs one at a time.
Mix in the crumbled Stilton and season with salt and pepper.
Fry teaspoon-sized batter portions for 3 minutes until golden.
Drain the beignets on kitchen paper and season with salt.
Whisk the vinegar and oil together in a bowl.
Blend the remaining Stilton and double cream in a processor until smooth.
Toss the beetroots, shallots, beignets, and lettuce with the oil dressing.
Serve in bowls topped with more beignets and a quenelle of Stilton dressing.