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.
Cook's Note: When blending hot liquids, remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.
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.
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.
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.