Pressure cook the toor dal and keep aside.
Cut the vegetables into medium size chunks. This could be done the night before to ease the process.
Soak the tamarind in water and squeeze the juice. Combine the vegetables with the tamarind juice and add the Sāmbhar powder and asafetida. Do not add salt at this point. Let the vegetables cook well and then add the salt.
Meanwhile, fry all the ingredients mentioned and grind it into a smooth paste.
Let the Sāmbhar boil well and check the vegetables, if they are cooked. This could take at least 45 mins, since certain root vegetables take longer to cook. Once the vegetables are cooked, add the salt and let it boil once.
Now add the ground mixture and the cooked dal and let it come to boil. Let it simmer for 15-20mins and then add the jaggery.
Season it with mustard seeds, urad dal, asafetida, and curry leaves. Serve hot with kali.