1. Place the pigeon peas in a small saucepan. Fill the pan halfway with water and rinse the peas by rubbing them between your fingertips. The water will become cloudy. Drain this water. Repeat three or four times, until the water remains relatively clear; drain. Now add 2 cups water and bring it to a boil, uncovered, over medium-high heat. Skim off and discard any foam that forms on the surface. Reduce the heat to medium-low, cover the pan, and simmer, stirring occasionally, until the peas are tender, about 20 minutes.
2. While the pigeon peas are cooking, pour 4 cups water into a medium-size saucepan. If you are using the ball of tamarind, add it to the water and allow it to soften, 15 to 20 minutes. Then, using your fingers, break it apart and coax the pulp to leach and dissolve into the water as you massage the pieces. Once the water is light chocolate-brown and tastes tart, strain it through a fine-mesh strainer to separate out the pulp and any fibers; discard the pulp and fibers. Reserve the tamarind-infused water. If you are using tamarind paste, simply whisk it into the water to dissolve it.
3. Sprinkle the Sambhar masala and asafetida over the tamarind water. Add the cauliflower, pumpkin, “drumsticks,” and curry leaves. Heat to a boil. Then reduce the heat to medium and simmer the thin brothlike curry, uncovered, stirring occasionally, until the vegetables are fork-tender, 15 to 20 minutes.
4. When the pigeon peas are tender, set the saucepan aside.
5. Heat 1 tablespoon of the oil in a small skillet over medium-high heat. Add the cashews, coriander seeds, and chiles, and stir-fry until the nuts and seeds turn reddish brown and the chiles blacken, 1 to 2 minutes. Scrape the skillet’s contents into a blender jar, and add ½ cup water and the coconut. Puree, scraping the inside of the jar as needed, to form a slightly gritty paste. Add this paste to the pumpkin mixture. Pour ½ cup water into the blender jar, swish it around to wash the jar out, and add this to the pumpkin mixture.
6. Heat the remaining 1 tablespoon oil in the same skillet over medium-high heat. Add the mustard seeds, cover the skillet, and cook until the seeds have stopped popping (not unlike popcorn), about 30 seconds. Add this spiced oil to the pumpkin mixture.
7. Transfer the cooked pigeon peas, with their cooking water, to a blender and puree, scraping the inside of the jar as needed, until smooth. (If you have an immersion blender, you can puree the peas and water right in the saucepan.) Pour this thin, creamy-yellow broth into the pumpkin mixture, and add the salt. Bring to a boil and then reduce the heat. Simmer, uncovered, stirring occasionally, until the flavors blend, about 5 minutes. Then serve.