Make the mussels: Bring water, wine, and garlic to a simmer in a large skillet. Add mussels, and cover tightly. Cook, about 2 minutes. With a slotted spoon, transfer mussels to a bowl as they open. Discard any unopened mussels. Refrigerate until chilled, at least 30 minutes.
Meanwhile, make the saffron mayonnaise: Simmer garlic in a small saucepan of water until tender, 15 minutes. Drain, and let cool. Crush saffron threads and 3/4 teaspoon salt with a mortar and pestle or with the back of a spoon. Puree garlic, yolks, and saffron salt in a food processor, or in a bowl using an immersion blender. With machine running, add vegetable oil in a slow, steady trickle, until mixture begins to emulsify. Continue adding remaining vegetable oil slowly, processing until incorporated. Gradually add olive oil, vinegar, and water, blending until silky. Transfer to a small bowl. (Mayonnaise can be refrigerated for up to 1 day.) Stir before serving.
Remove mussels from their shells, and return each one to a half shell. Spoon mayonnaise on top of each mussel, and serve with vegetables on the side and remaining mayonnaise for dipping.
The eggs in this recipe are not cooked. This dish should not be prepared for pregnant women, babies, young children, the elderly, or anyone whose health is compromised.