In a large pot, combine peas, broth, water, and soup mix. Stir well. Add carrots, onions, celery, leeks, garlic, mustard, tamari, and balsamic vinegar. If mixture seems too thick, pour in a little more water.
Bring to a full boil over medium, reduce heat to a simmer, and cook for about 2 hours, stirring and skimming surface occasionally.
Remove from heat and let sit for an hour at room temperature.
Puree, in several batches, in a food processor or blender until smooth. Add salt and pepper to taste.