Bouillabaisse

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1. 1Step 1: In a large pot over medium heat, add olive oil, fish carcasses, shrimp heads, and shells, and tomato paste. Stir until the shells are dark red and the fish starts to stick. Transfer the contents to a bowl and set aside. Deglaze the pot with the wine, scraping well the bottom of the pan. Add the wine to the bowl with the fish and shrimp.
2. 2Step 2: In the same pot, add more oil and cook the onions until translucent. Add back the fish and shrimp mixture with the wine, along with the remaining ingredients, including the celery leaves and parsley stems. Add enough water to barely cover and simmer for 45 minutes. Strain through a fine sieve, cheesecloth, or a chinois. Save a few carrot slices per person (carrots should not be very soft).
3. 3Step 3: Keep the strained stock simmering; reduce it if not flavorful enough. Season with salt.
4. 4Step 4: Cube some potatoes, boil in salted water for a few minutes, then finish cooking in a little bit of fish stock.
5. 5Step 5: For each person, count 1/2 fish fillet (skin removed), 3 mussels, 3 shrimp, 3 cherry tomatoes, and 3 cubes (or balls, if using a melon baller) of cooked potatoes.
6. 6Step 6: Bring the stock to a simmer. Poach the shrimp and set aside in a warm bowl (about 4 minutes). Poach the mussels and set aside (about 2 minutes).
7. 7Step 7: Poach the halved fish fillets (about 5 minutes).
8. 8Step 8: In serving bowls, put carrot slices, cherry tomatoes, and cooked potatoes. Cover with stock. Add the cooked mussels, shrimp, and fish. Sprinkle with parsley.
9. 9Step 9: If desired, serve with baguette slices topped with garlic mayonnaise.