Heat oil in a pan over medium-high heat.
Once the oil is hot, add urad dal, chana dal, asafetida, cumin seeds, and dry red chilies and saute until they turn slightly brown (50-60 seconds), stirring frequently.
Add cabbage to the pan and mix well. Reduce the heat to medium-low.
Cover the pan with a lid and cook for 6-8 minutes, stirring a few times while cooking.
Once the cabbage is cooked, remove the lid of the pan.
Add salt and tamarind paste to the pan and saute for a minute. Remove the pan from the heat and let the mixture cool down completely.
Transfer the mixture to the small jar of a blender along with ¼ cup water and grind to make a smooth chutney. Scrape the sides of the blender a few times while blending.
Add some more water if the pachadi is thick for your liking. Check for salt and tamarind paste and add more if required. Transfer the chutney to a serving bowl.
Heat oil in a small pan. Once the oil is hot, add mustard seeds, curry leaves, and dry red chilies, and let them crackle for 4-5 seconds.
Pour the tempering over the chutney and mix well. Serve.