In a medium, deep skillet with a tight-fitting lid, combine the sunchokes with 1 tablespoon of the butter and stir over moderately high heat until the butter is melted.
Add 1/2 cup of water, season with salt and pepper and bring to a boil.
Cover and cook over moderate heat until the liquid is evaporated and the sunchokes are tender and lightly browned, 8 to 10 minutes.
Transfer the sunchokes to a bowl and wipe out the skillet.
Fill the skillet with water and bring to a simmer.
In a large saucepan, combine the milk with 1 cup of water, season with salt and bring to a boil.
Whisk in the polenta and cook over moderate heat, whisking, until tender, about 5 minutes.
Stir in the 1 1/2 cups of shredded cheese and the remaining 2 tablespoons of butter until creamy; stir in a few tablespoons of water if the polenta is too stiff.
Crack the eggs into ramekins and gently pour them into the simmering water in the skillet, one at a time.
Poach the eggs until the whites are set and the yolks are runny, about 4 minutes.
Using a slotted spoon, transfer the eggs to a paper towellined plate.
Spoon the polenta and sunchokes into shallow bowls and top with the eggs.
Garnish with chervil and cheese, sprinkle with truffle salt and serve right away.