Place potatoes in a 4-quart Dutch oven and add enough lightly salted water to cover. Bring to boiling; reduce heat. Simmer, covered, for 20 to 25 minutes or until potatoes are just tender. Drain well; cool slightly. Peel and cube the potatoes.
Meanwhile, combine tofu, lemon juice, mustard, garlic and 1 teaspoon salt in a food processor or blender. Process or blend until smooth. With the processor or blender running, add oil in a thin, steady stream. When necessary, stop processor or blender and scrape down sides. Set aside.
In a very large bowl, combine cubed potatoes, celery, red pepper, eggs, onions and pickles. Add dressing, tossing lightly to coat. Season to taste with additional salt and pepper. Cover and chill for 4 to 24 hours. Before serving, add milk or pickle juice, 1 tablespoon at a time, to reach desired consistency.