Heat a large pot over high heat. Add olive oil and 1/2 teaspoon thyme. And onion and bacon and sauté until onion is tender. Add garlic and 1/2 teaspoon salt. Cook about 1 minute. Add greens a little at a time, cooking until wilted. Add 1/2 cup chicken broth. Once the greens are cooked down, add balsamic vinegar. Cover and simmer 5 minutes.
Gently stir in peas, tomatoes, kielbasa, water, remaining 31/2 cups chicken broth, remaining1/2 teaspoon thyme, remaining1/2 teaspoon salt and black pepper. Stir well. Cover and simmer over low heat until liquid is reduced by one-third, about 45 minutes. Uncover and continue to simmer until desired consistency is reached. If using canned black-eyed peas, rinse and drain the peas, then add to the pot during the last 15 minutes of cooking.