For the salad: Bring a large pot of water to a boil over medium heat. Add the potatoes and boil until tender, about 25 minutes. Drain the water and let cool. Quarter the potatoes and transfer them to a large serving bowl.
In a large skillet, heat the 2 teaspoons of oil over medium-high heat. Add the kielbasa slices and cook until brown. Transfer them to a paper towel lined plate to drain. Add the kielbasa to the bowl with the potatoes, along with the celery and green onions.
For the dressing: In a small bowl, whisk together the vinegar, mustard, garlic, salt, and pepper. Slowly whisk in the oil in a long steady stream. Add the dressing to the potato mixture and toss to combine. Serve warm or at room temperature.
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