In a small pot over medium heat, add the jaggery and 1½ cups water. Cook, stirring, until the jaggery dissolves.
Strain the mixture through a fine sieve, then set aside to cool slightly.
In a food processor, process the bananas until very smooth.
In a large bowl, add the rice flour, coconut, cardamom, ginger, and baking soda; stir to combine. Add the banana purée and the jaggery syrup and stir until a loose, wet batter forms. Cover and let rest for 3–4 hours.
Set a paper towel–lined plate next to the stove. In an appachatti pan over high heat, distribute ½ cup melted ghee evenly among the divots. When the ghee is nearly smoking (about 2 minutes), lower the heat to medium and carefully but quickly drop 1 tablespoon batter into each divot. Cook until browned and crispy on one side, about 2 minutes, then flip and cook 2–3 minutes more.
Transfer to the prepared plate and continue cooking the batter in batches, refilling the divots with the remaining ¼ cup ghee as needed. Serve the unniyappam warm.