While rice is cooking in skillet, combine ghee, garlic, and chile in reserved pot and place over medium heat. As soon as oil around garlic starts to sizzle, start stirring occasionally and cook until garlic is barely golden, about 3 minutes. Sprinkle in coriander, cumin seeds, and turmeric and mix until turmeric has stained garlic and ghee, about 20 seconds. Scrape in tomato paste and cook, stirring often, until slightly darkened in color and beginning to stick to bottom of pot, about 3 minutes. Pour in 5 cups cold water; bring to a boil. Reduce heat to medium-low; add sugar and lime juice and season with salt. The broth will be thin but should have lots of flavor. Taste and add more lime juice and/or salt if needed.