In a bowl, combine boiled potatoes, black chickpeas, roasted cumin, chaat masala, red chili powder, black salt, and fresh cilantro. Mash coarsely with your hands so it remains textured.
Add mint, cilantro, green chilies, and ginger into a blender with 1/4 cup of ice-cold water. Blend until completely smooth and bright green.
Combine the green paste with 4 cups of chilled water, tamarind paste, black salt, chaat masala, and roasted cumin. Strain through a fine-mesh sieve into a serving bowl.
In another bowl, mix 2 cups of water with tamarind paste, jaggery, black salt, and roasted cumin powder until the sugar dissolves completely. Chill.
Heat oil in a deep pan. Fry ready-to-fry pellets a few at a time until they puff up fully into crispy spheres. Drain on paper towels.
Poke a hole in a puri, stuff it with a spoonful of potato filling, drizzle a little sweet water, fill it to the brim with spicy water, and eat it immediately in one bite.