Grind the coconut, cumin seeds and green chillies into a fine paste. Keep aside. Whisk the yogurt till smooth and keep aside.
Cut the vegetables into pieces about 1 1/2 inches in length (like we cut potatoes for finger chips). Cook the vegetables with turmeric powder and half a cup water till soft but firm. I cook my vegetables in the microwave (without turmeric powder) at 100% for about 8 minutes.
You can also cook this on the stove top till the vegetables are done.
Then I put the vegetables into a pan with 1/8th cup of water, curry leaves, salt and turmeric powder. When this comes to a boil, add the coconut paste and mix well. Turn down the heat and add the yogurt. Mix well.
When the mixture just starts to bubble, take off the heat. The aviyal will have very little liquid in it and the coconut paste-yogurt mixture should be thickly coating the vegetables. Pour the coconut oil and stir. Cover.
Mix in the coconut oil just before serving. Serve warm or at room temperature with rice, sambhar, pappads and pickle.