Heat 2 tablespoons butter in a large pot over medium-high heat. Add the onion, carrots, a big pinch of salt and a few grinds of pepper. Cook, stirring occasionally, until the onion is translucent, about 4 minutes. Stir in the ginger and curry powder and cook, stirring, until the spices are toasted, about 30 seconds. Stir in the tomatoes and cook until softened, about 2 minutes. Add the chicken broth, 3 cups water, the chard stems, 1/2 teaspoon salt and a few grinds of pepper. Cover and bring to a boil over high heat.
Meanwhile, heat the frozen rice in the microwave as the label directs. Cut the paneer into 1/2-inch cubes. Heat the remaining 1 tablespoon butter in a medium nonstick skillet over medium-high heat. Add the paneer and cook, turning with a small spoon, until well browned on a few sides, 3 to 5 minutes (reduce the heat if necessary). Transfer to a plate and sprinkle with salt.
Uncover the soup and reduce the heat to a simmer. Add the rice and chicken; simmer until the carrots are tender, 5 minutes. Stir in the chard leaves and simmer until tender, 5 minutes. Add more water if needed and season with salt and pepper.