Melt the butter in a large saute pan over medium-high heat. Add the shallots and cook until translucent, about 3 minutes. Add the shrimp and saute until cooked through, about 3 minutes. Add the chives and lemon juice, and season with salt and pepper. Place a spoonful of the Bouillabaisse Sauce on a plate. Place a piece of baguette on top of the sauce, and then a large spoonful of the cooked shrimp. Top with the Bouillabaisse Relish. Finish with a spoonful of the Saffron Aioli.
Heat the oil in a large saucepan over medium-high heat. Add the fennel, onions and garlic and cook, stirring frequently to prevent sticking, for about 10 minutes. Then add the tomatoes, peppers and thyme. Continue to cook and stir for 8 minutes. Add the white wine and cook until it has reduced by half. Then add the fish stock. Increase the heat and bring to a boil. Add the fish and decrease the heat to a simmer. Cook for 5 minutes. Add the basil and Espelette, then turn off the heat and let cool slightly. Place the sauce into a blender and blend until smooth. Season with salt and pepper.
Heat the olive oil in a small skillet over medium heat and saute the carrot, celery, fennel and onions until soft, about 5 minutes. Add the basil. Season with salt and pepper.
In small saucepan, heat the saffron threads and 1 tablespoon water to a boil and set aside to cool. Put the garlic in a small bowl with the egg and some pepper and whisk until combined. Slowly whisk in the olive oil, in drops at first and increasing the amount added as the sauce thickens. Add the lemon juice, saffron water and season with salt and pepper.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.