Combine the garlic, thyme and 1 tablespoon olive oil in a nonstick skillet over medium heat and cook until fragrant, about 2 minutes. Add the black-eyed peas and 2 tablespoons pickling liquid and cook until tender, about 10 minutes. Add the turkey and heat through, about 3 more minutes. Remove from the heat and add the scallions, and salt and cayenne to taste. Discard the thyme.
Meanwhile, whisk the mustard and the remaining 2 tablespoons pickling liquid in a large bowl. Add salt and cayenne to taste, then gradually whisk in the remaining 3 tablespoons olive oil. Add the tomatoes and toss. Add the arugula, celery and pickles and toss again. Divide among bowls and top with the black-eyed peas and turkey.