Whisk the chickpea flour, yogurt, ginger, turmeric, pepper and salt together. Whisk in water until the mixture is thick but pourable, like pancake batter. Set the mixture aside for a few hours or overnight at room temperature until it ferments (the batter will rise and there will be bubbles on top).
Heat a seasoned skillet over medium heat and add the oil. Pour about 1/2 cup of the batter into the skillet and spread it with the back of a ladle into an even circle. (Save the remaining batter for another use.) Cook until the edges are golden; flip the chilla gently and cook until the other side is browned. Stuff with the carrots and beets and eggs if using.