Bring a large pot of water to a boil. Add the potatoes; boil about 25 minutes or until tender. Drain and let cool. Cut potatoes into quarters and place in a large bowl.
In a large skillet, heat the 2 teaspoons oil over medium-high heat. Brown the kielbasa. Transfer to a paper towel-lined plate to drain. Add kielbasa to the bowl with the potatoes along with celery and green onions.
For the dressing, in a small bowl, whisk together the vinegar, mustard, garlic, salt, and pepper. Slowly whisk in oil. Add the dressing, celery and green onions to the potato mixture and toss to combine. Serve warm or at room temperature.