Bring a large pot of salted water to a rolling boil over medium-high heat. Add potatoes and cook until just fork tender. Drain potatoes into a colander, allowing the steam to evaporate for 5–10 minutes.
Combine the next 7 ingredients into the bottom of a large mixing bowl and whisk to make an emulsion.
Add potatoes along with the remaining ingredients and stir until ingredients are blended and potatoes are coated in the vinaigrette. Serve warm or at room temperature garnished with za’atar.