Place the mung beans and brown rice in a medium bowl and cover with water by 3 inches. Let soak overnight or for 8 hours. The next day, drain the beans and rice and set aside.
In a large soup pot, toast the cumin and mustard seeds over medium heat for 3 to 4 minutes until they become fragrant and you start hearing them pop.
Add the onion and carrot and a few tablespoons of water and increase the heat. The water and onions will sizzle since the pot is already hot. Sauté for 5 to 7 minutes until the onions begin to soften, adding water as needed to prevent burning.
Next add the garlic, ginger, tomato, garam masala, coriander, turmeric, salt, and pepper and cook for another 30 seconds. Add the mung beans and rice and cook for another minute, stirring.
Next, add the broth and bring to a boil. Once boiling, reduce to a simmer. Cover and simmer over low heat for 45 to 50 minutes, stirring occasionally, until the beans and rice are very tender and cooked through.
Remove the kitchari from heat and let rest for a few minutes. If you want a looser, more brothy kitchari, you can add more broth. I like a thicker kitchari and let it rest covered for 10 minutes so all the broth gets absorbed before spooning into bowls! Taste and reseason as needed. Serve topped with cilantro, lemon wedge, and diced avocado (if desired).