In a 12-inch frying pan or 5- to 6-quart pan, combine onions and oil; cook over medium-high heat, stirring frequently, until onions are limp, 10 to 15 minutes.
Reduce heat to medium-low and continue cooking, stirring occasionally, until onions are golden and sweet tasting, 15 to 30 minutes longer.
Let onions cool.
Shortly before serving, stir together cooked onions, yogurt, vinegar, and salt to taste.
Spoon onion mixture into a serving bowl.
Cover mixture generously with pepper.