Heat oil in a large heavy pot over medium-high heat. Add onions; season with salt and pepper. Cook, stirring occasionally, until onions are soft and light golden brown, about 10 minutes. Add leeks, next 5 ingredients, and 1 teaspoon salt. Cook, stirring often, for 5 minutes. Add wine; cook until reduced by half, about 2 minutes. Add stock. Bring to a boil, reduce heat, and gently simmer for 15 minutes.
Season broth to taste with salt and pepper. Remove from heat; discard thyme sprigs and bay leaves. DO AHEAD Broth can be made 1 day ahead. Let cool slightly; chill, uncovered, until cold. Cover; keep chilled.
Purée first 4 ingredients and 1/8 teaspoon cayenne in a food processor, occasionally scraping down sides of bowl, until a smooth paste forms. With motor running, slowly drizzle in oil until a thick mayonnaise-like emulsion forms. Transfer to a medium bowl. Whisk in 1 tablespoon lemon juice; season with salt, pepper, and more cayenne and lemon juice, if desired. Cover; chill for at least 1 hour while flavors meld. DO AHEAD Rouille can be made 3 hours ahead. Keep chilled.
If chilled, rewarm bouillabaisse broth; cover and set aside.
Meanwhile, place oil in a pot large enough to hold all mussels; heat over medium-high heat. Add garlic and red pepper flakes; cook, stirring constantly, until sizzling but not brown, 10–15 seconds. Add mussels, increase heat to high, and stir until evenly coated. Add Pernod; cover pot. Steam mussels, stirring once or twice, until they open, 7–10 minutes (depending on size of pot; discard any mussels that do not open). Stir in parsley and hot bouillabaisse broth. Ladle mussels and broth into bowls. Serve with rouille and garlic toast (rouille can be added to broth and spread on toast).
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