In a large wide stainless-steel pot, combine the water, the 2 tablespoons oil, 1 1/2 tablespoons of the vinegar, the onion, peppercorns, bay leaves, and 1/4 teaspoon of the salt. Bring to a boil.
Cut the stems off the artichokes and cut off the top third of the leaves. If you like, using scissors, cut off the tips from the remaining leaves to remove the thorns. Put the artichokes in the pot, stem ends down. Cover and bring back to a boil. Reduce the heat and simmer until the bases of the artichokes are tender when pierced with a small knife, 30 to 40 minutes. Remove the artichokes.
In a small glass or stainless-steel bowl, whisk together the mustard, the remaining 4 tablespoons vinegar, the scallions, parsley, lemon juice, the remaining 1 1/4 teaspoons salt, and the ground pepper. Add the remaining 1 cup oil slowly, whisking. Serve each artichoke with a small bowl of vinaigrette alongside for dipping.