Heat oil in a skillet over medium-high heat. Stir curry paste into oil until smooth. Add chicken; stir to coat each piece.
Pour 1 can coconut milk into skillet and stir until curry paste is completely dissolved and begins to bubble, 3 to 5 minutes. Pour remaining coconut milk, chicken stock, onion, potatoes, and chile peppers into curry paste mixture. Bring to a boil, reduce heat to low, and simmer until potatoes are tender, about 25 minutes. Remove from heat.
Stir fish sauce into chicken curry mixture just before serving.
