Rinse ⅓ cup Carolina Gold or other parboiled rice, ⅓ cup toor dal (pigeon peas), ⅓ cup urad dal, and ⅓ cup yellow split peas in a fine-mesh sieve until water runs mostly clear. Transfer to a medium bowl, pour in 3 cups hot water to cover, and let soak at least 2 hours and up to 4 hours.

Transfer rice mixture, including water, to a blender; reserve bowl. Add one 1" piece ginger, peeled, coarsely chopped, 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt, 2 tsp. ground asafetida, 1 tsp. cumin seeds, 1 tsp. Kashmiri chile powder or Hungarian hot paprika, and ¼ tsp. ground turmeric and blend until very smooth, about 5 minutes. Pour batter into reserved bowl. The consistency should be similar to heavy cream. Add water a few tablespoons at a time to adjust consistency if needed.

Heat a medium nonstick skillet over medium. Lightly brush skillet with melted ghee or clarified butter. Ladle ¼ cup batter into pan and quickly spread batter outward in a circular motion with ladle to make a 6"–7"-diameter circle. Sprinkle some finely chopped white onion and finely chopped cilantro on top if desired. Drizzle ½ tsp. melted ghee or clarified butter over. Cook until outer edges of adai begin to look dry, about 2 minutes. Using a heatproof rubber spatula, carefully loosen adai (it should be crisp and brown underneath) and fold one half up and over so ends meet to make a semicircle. Transfer to a wire rack set inside a rimmed baking sheet. Repeat with remaining batter and remaining 5½–6½ tsp. melted ghee or clarified butter and more onion and cilantro if desired (you should end up with 12–14 adai total). Do Ahead: Batter can be made 1 week ahead. Transfer to an airtight container; cover and chill. Bring to room temperature before using.