Heat oil in a pot over medium heat. Add the Massaman curry paste and sauté until fragrant. Stir in coconut milk, then add the chicken drumsticks or thighs. Cook the chicken with the curry paste until combined. Add more coconut milk and water.

Add potatoes, carrots, star anise, cinnamon stick, and roasted peanuts. Cover the pot.

When the mixture comes to a boil again, season with palm sugar and salt. Cover and reduce to medium heat. Simmer until the chicken is tender, about 1 hour and 30 minutes. Taste and adjust seasoning as needed until you reach your desired flavor.