Soak the black eyed peas in water for 30 minutes.
Bring the vegetable broth to a boil in a medium saucepan over high heat. Add the black eyed peas, bay leaf, thyme and garlic, return to a simmer and cook for 1 hour, until tender. Check after 45 minutes and add more broth or water as necessary to keep them submerged.
Drain the black eyed peas well and transfer to a clean dish towel to dry.
Heat 1 inch of canola oil in a skillet to 325°F. When the oil is hot, fry the black eyed peas until crisp and golden, working in batches if necessary. Drain the fried peas on a paper-towel lined plate and toss with salt and Creole seasoning as soon as they come out of the oil.
Bring a large pot of salted water to a simmer. When simmering, add the shrimp and cook until just pink, about 3 minutes. Immediately remove the shrimp with a strainer (keep the pot of water on the stove) and run under cold water for several minutes.
Return the pot of water to a boil for the eggs. Gently place the eggs in the boiling water and cook at a high simmer for 7 minutes. Immediately remove the eggs from the water and run under cold water for several minutes. When cool, peel the eggs. For a more firm yolk, cook the eggs for up to 12 minutes.
To make the vinaigrette, combine all of the ingredients except the oil in a blender, food processor, or immersion blender. As the mixture is blending, slowly add the oil to emulsify. Season to taste with salt and pepper.
Place all of the remoulade ingredients in a blender or bowl with an immersion blender. Purée until smooth and adjust the seasoning to taste.
Toss the shrimp with the remoulade sauce until well coated.
In a large bowl, toss the lettuce, onion, cucumber and radishes with the red wine vinaigrette. Divide between plates and top with remoulade shrimp, crispy black eyed peas, tarragon, parsley and chives.