Melt the coconut oil. Add the mustard seeds and cumin seeds and toast them until the mustard seeds start to pop. Add the diced onions, diced carrots, minced ginger, and garlic. Cook for about 5 minutes, until onions have softened.
Once the onions have softened, add the salt, turmeric, cloves, and black pepper and stir until the spices coat the vegetables. Add the rice, moong dal, vegetable broth, and water. Bring everything to a boil. Cover the pot and simmer over low heat for about 20 minutes.
If the kitchari is looking too runny, let it simmer, uncovered, for a few more minutes. If the kitchari is too thick, turn off the heat and add 1/2 to 1 cup of water and stir.
Serve in bowls, along with chili, cilantro, and lemon if desired.