Rinse rice in several changes of water and transfer to a container. Cover with water and let soak at room temperature, 15 minutes to 1 hour. Drain.

Combine 3 cups water with red lentils in a large pot over medium heat. Bring to a simmer, skimming any scum that collects on the surface. Stir in onion, cumin, coriander, salt, black pepper, cayenne, and turmeric. Reduce heat to low and simmer, partially covered, until lentils are tender, about 30 minutes.

Stir broccoli, tomatoes, peas, and salt into the pot. Continue cooking, stirring frequently, until lentils are soft, about 30 minutes more.

Combine drained rice with remaining 1 1/2 cup water in another pot; bring to a boil. Reduce heat to very low and simmer, covered with a tight-fitting lid, until rice is tender and liquid is absorbed, about 20 minutes.

Heat oil in a small skillet over medium heat. Add mushrooms, ginger, and garlic; cook and stir until fragrant, 3 to 5 minutes. Pour over lentils; stir to distribute. Cook lentils until flavors combine, about 5 minutes. Serve lentils with rice.