Place chicken into a large saucepan and cover with water; bring to a boil. Reduce heat to medium and simmer until chicken is no longer pink in the center, about 20 minutes.

Meanwhile, heat vegetable oil in a large skillet over medium heat; cook and stir onions and garlic until onion is tender, about 5 minutes. Stir mild curry powder, hot curry powder, black pepper, and salt into onion mixture; continue to cook and stir for 5 minutes, then transfer mixture to pan with chicken.

Stir potatoes into chicken mixture. Simmer until potatoes are tender, about 20 minutes more.