Add 1/2 cup boiling water to the tamarind in a medium bowl. Cover and keep aside for 10 minutes. Squeeze out all the extract from the tamarind into the water in which it was soaked. Squeeze the tamarind dry. Sieve or pass the tamarind extract between fingers to collect and discard any fiber/seeds. Collect extract into a cup and keep aside. You'll get 3/4 cup extract from this process.
Wash well and pressure cook the toor dal with little over a cup of water, with a pinch of turmeric powder. Since this is carrot sambar, pressure cook sliced carrots with 1/4 cup water and pinch of salt in another vessel and cook them by stacking the vessels one of top of the other to save time. After 3 whistles, reduce flame to minimum and keep on sim for 12 minutes or so. Remove dal and carrots when cooker has cooled. Mash the dal and keep aside. Drain water from the carrots and keep aside to use if required later.
While the dal is cooking in the pressure cooker, in a kadai, using a few drops of oil, toast the red chillies, chana dal, asafoetida chunk, coriander seeds, curry leaves and fenugreek seeds, on a medium-low flame until the dal is golden, chillies are bright and asafoetida has puffed up. Take care not to burn the fenugreek seeds.
You can add the desiccated coconut at this stage and toast them altogether until the coconut is golden-brown. Coconut is entirely optional. Most of the times I omit the coconut but in some tambrahm families, arachuvitta sambar without coconut is sacrilege. Roasted coconut lends an intense aroma to the sambar. If you don't have desiccated coconut, you could use fresh coconut and roast it similarly until it is golden.
Once cooled, place all the masala ingredients in the small jar of a mixer and grind till you get a fine powder. You can add up to 1/4 cup of water until you get a smooth paste and then add 3-4 tbsp of water to clean up the mixer jar. Collect all this masala in a bowl and keep aside.
At this point of time, the masala, tamarind extract, carrots and dal are ready and it's time to prepare the sambar.
In a kadai, combine the carrots and only 1/2 cup of tamarind extract. Keep the remaining aside to be added later if you think that sambar needs more souring. Bring this to a simmer for 2 minutes or so. To this add the ground masala and bring to a simmer.
When this is simmering, add the mashed dal, 1 tsp salt. Bring this to a gentle simmer. If you think the sambar is too thick, add the water kept aside from boiling the carrots and bring to a simmer. At this stage, check for salt and remove into a serving bowl.
In a tempering ladle, heat 1-2 tsp of oil. When the oil is hot, add the mustard seeds, red chilli and curry leaves. When the seeds splutter, add the asafoetida and switch off the flame. Add this to the sambar and cover the serving bowl immediately to seal all the flavours of the tempering.