To make the aioli: Place the red pepper on an open flame or grill. Turn to blacken on all sides. When fully charred, remove to a bowl and cover with plastic wrap. Let pepper cool, until you can handle it comfortably. Using your fingers, remove blackened skin to reveal the red color underneath. (A paper towel will wipe off what you can't remove with your fingers.) Coarsely chop the pepper and place it in the bowl of a food processor. Add the yolks, garlic, vinegar, chili paste, and salt to taste. Puree until the mixture is homogeneous. While the machine is running VERY slowly drizzle in the oil, until incorporated. Check the aioli for texture and flavor. If it's too thick, add a few drops of water to thin it down, or on the flip side if it's too thin add more oil. The desired outcome is a fairly thin aioli. Check the flavor and add more salt or chili paste, if desired. Refrigerate until ready to use.
To cook the mussels: In a pot, large enough to accommodate all the mussels, generously coat with olive oil. Add the onions and bring to a medium-high heat, stirring frequently. Cook the onions until they are very soft and aromatic but have no color. Add the garlic, thyme and bay leaf and cook for another 2 to3 minutes. Stir to be sure the garlic doesn't burn. Add the mussels and stir to coat with the oil. Add the wine and cover the pot. Steam the mussels for 5 to 6 minutes or until all the shells open. If some shells don't open, cook them for a little longer. Sometimes the freshest mussels are the most stubborn to open. Remove the mussels that don't open after longer cooking. When in doubt, throw out.
While the mussels are steaming, grill or toast the slices of bread.
Serve the mussels in individual bowls with lots of the juice from the bottom of the pot. Generously drizzle each bowl with the aioli. Garnish with toast. (You may want to have more toasted bread on hand to sop up the delicious liquid!)
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight 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.