Prepare the aoli: Mix the bread crumbs with the vinegar. When absorbed, wring out the excess liquid in paper toweling.
Using the metal blade of a food processor, combine the bread crumb mixture, egg yolks, garlic, salt, and pepper. Pulse on and off until smooth. With the machine running, gradually add the olive oil in a slow steady stream and process until the aoli is very thick. Add the lemon juice and 1 tablespoon of boiling water to lighten the mixture.
Using a rubber spatula, scrape the mixture into a bowl, cover, and refrigerate until ready to use. (The aoli may be prepared 1 day ahead.)
Prepare the vegetables: Place the fennel wedges in a large saut pan and add enough chicken stock to barely cover. Fit the pan with a buttered round of waxed paper. Bring the liquid to a simmer over medium heat. Cook the fennel for 20 minutes until fork tender. Set the fennel aside in the cooking liquid to keep warm.
Boil the potatoes in a large saucepan of lightly salted water until tender, about 15 minutes. Drain well. Wrap the potatoes in a clean kitchen towel, return to the pot, and cover to keep warm.
Cut the artichokes half lengthwise and rub them with the lemon. Using a small paring knife, scrape away the thistley chokes. In a nonreactive pan, heat the oil over medium heat, add the artichokes, cover and, shaking the pan occasionally, saut until tender, about 10 minutes. Season with salt and pepper. Cover to keep warm and set aside.
Prepare the bourride: In a soup kettle, add the fish stock and bring to a simmer over medium heat. Rinse the fillets in cold water, drain, and pat dry. Add to the stock and simmer for 2 to 3 minutes until the fish is barely opaque. Place half of the aoli in a medium bowl and gradually whisk in a ladle or two of the fish stock. Stir the aoli-stock back into the kettle and cook another minute until slightly thickened. (Do not boil.)
Ladle the soup into warm soup plates, sprinkle with parsley, and serve with croutons, vegetables, and the remaining aoli on the side.
Note: For a summer lunch or starter, serve the aoli as a dipping sauce for a variety of raw vegetables, such as tomatoes, celery, sweet peppers, wax beans, baby carrots, and scallions. Accompany with hard-boiled eggs and boiled new potatoes.
