If using store-bought mayonnaise, stir the mayo, garlic, salt, lemon and pepper together in a small bowl. Set aside.
In a large saucepan or dutch oven, heat the olive oil over medium heat. Add the shallots and garlic, and cook, stirring, until soft and translucent, about 3 minutes. Add the beer and bring to a boil. Stir in the lemon juice and mustard until smooth. Add the mussels, cover the pot and cook, undisturbed, until the mussels open, about 5 minutes.
Uncover and using a slotted spoon, transfer the mussels to a serving bowl. Discard any that won’t open. Add the butter and chili and stir until broth is melted and emulsified. Taste the broth and season with salt and pepper, as desired. Pour the broth over top of mussels. Sprinkle with the parsley and basil, and serve hot with the aïoli and toasted bread on the side.