Sweet-and-Sour Dal Bhat

Bring tomato, toor dal, salt, and 8 cups water to a boil in a medium saucepan over medium-high heat, skimming foam from surface and stirring occasionally, 8–10 minutes. Reduce heat and simmer, stirring and skimming occasionally, until dal are very soft but not disintegrated, 45–60 minutes.
Meanwhile, process chiles and ginger in a food processor to a coarse purée, about 1 minute. Transfer to a 2-cup measuring glass or a medium bowl; stir in jaggery, lime juice, chile powder, and turmeric. Set seasoning mixture aside.
Once dal is done, mix in reserved seasoning mixture and simmer until flavors are melded, about 5 minutes. Blend dal with an immersion blender or in a blender until smooth; it should be about the consistency of heavy cream.
Do Ahead: Dal can be made 3–4 hours ahead. Cover and keep warm.
While the dal is cooking, place rice in a medium bowl and cover with cold water. Swish around with your fingers to remove surface starches, then drain through a sieve or colander. Repeat process until water runs clear, about 4 or 5 changes of water.
Heat 2 Tbsp. oil in a medium pot over medium. Add rice and stir to coat grains in oil. Add 4 cups water and bring to a boil over high; reduce heat to low, cover pot, and cook 15 minutes. Remove from heat.
Just before serving, heat remaining 7 Tbsp. oil in a small skillet over medium-high. Add curry leaves, chiles, mustard seeds, asafetida, and fenugreek one at a time, pausing for a second between each addition and swirling pan. Cook until spices are toasted and curry leaves sizzle and pop, about 20 seconds. Spoon spices over dal; serve with rice.