Preheat the oven to 190C/375F/Gas 5.
Mix the egg whites and table salt together in a large bowl until the mixture forms a paste.
Cover each beetroot in a 2cm/¾in thick layer of the salt paste, ensuring there are no gaps.
Place the salt-crusted beetroot on a baking tray and bake in the oven for 45-50 minutes.
Add the baby beetroot to a deep baking tray and add cold water to a depth of 2cm/¾in. Cover the tray tightly with foil and bake in the oven with the salt-crusted beetroot.
Remove from the oven and allow to cool.
Once cooled, crack the salt crust on the large beetroot and discard. Peel, then thinly slice half and chop the other half.
Peel the baby beetroot and quarter. Set aside.
Heat a deep fat fryer to 190C/375F. (CAUTION: Hot oil can be dangerous. Do not leave unattended)
Place 125ml/4fl oz water and the butter into a saucepan and bring to a rolling boil for a minute.
Remove from the heat and add the flour, beating well until the mixture is smooth and comes away from the sides of the pan.
Return to the heat for a minute or two to cook the flour out.
Remove from the heat again and add the eggs, one at a time, beating well between the addition of each egg.
Add the crumbled Stilton and mix well. Season with salt and freshly ground black pepper.
Drop teaspoonfuls of the batter into the deep fat fryer and fry for 2-3 minutes, or until golden-brown and crisp (you may need to do this in batches).
Remove from the fat with a slotted spoon and set aside to drain on kitchen paper. Season with a little salt.
In a separate small bowl, whisk together the vinegar and oil.
For the Stilton dressing, place the Stilton and double cream in a food processor and blend until smooth.
Place the quartered baby beetroots and shallots in a large bowl. Add the sliced and chopped salt-baked beetroot and almost all of the beignets along with the lettuce leaves and mix with the dressing.
Serve the salad in a wide serving bowls and top with the remaining beignets. Top with a quenelle of Stilton dressing.