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Place 2 tablespoons of the olive oil reserved from the artichokes in a medium saucepan and warm it over medium heat. Add the artichokes and cook, stirring occasionally, until they are lightly caramelized, about 5 minutes. Remove the pan from heat, transfer the artichokes to a plate, and let them cool.
Place the pan back over medium heat and add the mushrooms. Cover the pan and cook for 5 minutes. Remove the cover and cook until most of the liquid the mushrooms release has evaporated, about 5 minutes.
Add another tablespoon of the reserved oil and the leek and cook, stirring occasionally, until it has softened, about 5 minutes. Stir in the garlic, anchovies, and thyme and cook, stirring continually, for 1 minute.
Stir in the flour, cook for 1 minute, and then add the vermouth. Cook until the alcohol has been cooked off, 1 to 2 minutes.
While whisking, gradually add the stock. When all of the stock has been incorporated, add the celeriac and bay leaf along with the artichokes and bring the mixture to a boil. Reduce the heat and simmer until the celeriac is tender, 10 to 15 minutes, adding more stock if the level of liquid starts to look a bit too low.
Remove the pan from heat, remove the bay leaf, and discard it. Stir in the cream, tarragon, and vinegar, season the soup with salt and pepper, ladle it into warmed bowls, and enjoy.