Combine the mashed potatoes, salt, cayenne, turmeric, cilantro, and 3 tablespoons vegetable oil in a bowl and mix well. Add the whole wheat flour a little at a time, kneading the mixture until it forms a stiff but pliable dough. (If the dough becomes too stiff, add a few drops of water.) Transfer the dough to an oiled bowl, turning to coat; cover with plastic and set aside for 10 minutes.
Preheat a nonstick griddle to medium heat.
Divide the dough into lemon-sized portions and roll them out on a lightly floured surface to 1/4-inch thickness. Fry the chapatis using up to 1 teaspoon of oil for each side. Cook until brown spots have started to form on the bread and the dough is cooked through. Brush with melted butter and serve hot.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. The exact amount will vary depending on cooking time and temperature, ingredient density, and the specific type of oil used.