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.