Using a peeler and small utility knife, peel and remove both the outer skin of the celeriac and any folds that contain dirt or small pebbles. Then dice the celeriac into rough cubes and place in pot.
Wash the sunchokes thoroughly, making sure to remove any bits of dirt. Dice into pieces roughly the same size as the celeriac and place into the pot.
Add the shallots and cream. Pour in filtered water until the liquid covers over half an inch.
Turn on medium-high heat and simmer until root vegetables are tender.
Reserving the liquid, strain the vegetables and place them in a high-powered blender. Add the butter, vinegar, and reserved liquid, pulsing until silky smooth. Use a spoon to test (should completely coat spoon when dipped and removed from bisque). Add white pepper and salt to taste.
Serve bisque topped with tarragon oil, celery-compressed apple, frisée, and lavender.