Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain and rinse with cold water until cool enough to handle. Peel potatoes and cut into 1/2-inch cubes.
Heat oil in a skillet over medium heat. Cook and stir white and light green onion pieces until soft and slightly golden, 3 to 5 minutes. Add garlic, stir, and remove skillet from heat.
Stir potatoes, green onion-garlic mixture, dark green onion pieces, yogurt, yellow mustard, cayenne pepper, salt, and black pepper together in a bowl until evenly combined.