Step 1

In a saucepan, combine the potatoes and enough water to cover by 3 inches. Bring to a boil over high heat, reduce the heat to medium, and simmer until tender, 15 to 20 minutes, depending on size of potato. Drain, and let cool enough to handle, about 15 minutes. Using your fingers, peel off the skin. Place the potatoes in a medium bowl and mash them with a fork until they are mostly mashed but still slightly chunky (you will need about 1 3/4 cups mashed potatoes).

Step 2

In a medium sauté pan, heat 2 tablespoons of oil over medium-high until the oil glistens. Add the mustard seeds and cumin seeds and sauté until they begin to hiss, about 1 minute. Add the hing, curry leaves, turmeric, chile powder, ginger, and garlic. Sauté, stirring often, until the garlic is softened, about 3 minutes. Remove the pan from the heat. Season with a large pinch of salt and then add to the potatoes.

Step 3

Add the cilantro and chaat masala to the spice to the potato mixture. Stir with a wooden spoon, making sure to crumble the curry leaves, until everything is incorporated. Season with lime juice and salt to taste. Form the mixture into four 2-inch balls and arrange them on a plate in a single layer. Cover with a damp cloth and set aside at room temperature.

Step 4

In a large bowl, whisk together the chickpea flour and enough water, beginning with ¼ cup, and adding water more by tablespoonfuls to form a paste that resembles thick cake batter but is still thin enough to pour. It should not be runny. Stir in the coriander and season with salt.

Step 5

Line a plate with paper towels. Pour 4 inches of oil into a deep heavy-bottomed pot and heat to 350°F on an instant-read thermometer. Working in two batches, dredge each potato ball in the chickpea batter until it is well coated. Shake to remove any excess. Using a slotted spoon, gently lower the battered potato ball into the oil and fry until golden brown on all sides, 4 to 6 minutes, adjusting the heat to maintain 350°F. Turn the vada with the spoon as it fries to ensure even cooking and browning. Transfer to the paper towels to drain and season with salt. Repeat with the remaining vada.

Step 6

In a sauté pan, melt the butter over medium heat. Split the buns and place them in the pan, interior-side down; fry until light golden brown, about 1 minute. Place one bun, open and interior sides facing up, on each of four plates. Slather both sides of the bun with 2 to 3 teaspoons green chutney, then spoon about 2 teaspoons coconut chutney on top. Place a vada on the bottom bun and top with fried chiles (if using) and bun top. Serve with more chutney on the side.