Cut carrot, potato, and drumstick into long, thick batons.
Boil vegetables in water with turmeric powder and salt until cooked but firm.
Grind grated coconut, green chilies, and cumin seeds coarsely without adding much water.
Add coconut paste to cooked vegetables and simmer for 5 minutes.
Turn off heat, stir in whisked yogurt, fresh curry leaves, and a drizzle of raw vegetable oil.