Warm spices like ginger, mustard seeds, cumin, and coriander are a natural pairing with winter squash in this Indian-style curry that makes a satisfying dinner.
Puree onions, garlic, and ginger in a food processor until smooth. Heat a Dutch oven or other heavy-bottomed pot over medium-high. Add 1/4 cup oil and cumin and mustard seeds. Cook, stirring, until fragrant, 30 to 60 seconds.
Add onion mixture and cook, stirring occasionally, until beginning to brown, about 8 minutes. Reduce heat to low and cook, stirring frequently, until liquid has evaporated and bottom of pan begins to brown, about 3 minutes more. Stir in remaining 1 tablespoon oil and spices; cook 15 seconds. Stir in tomato paste and cook, stirring, 15 seconds more.
Add broth and cook, scraping up browned bits from bottom of pot. Add curry leaves and bring to a boil. Reduce to a simmer; cook, partially covered, 30 minutes.
Season chicken and squash with salt and add to pot, pushing down into liquid. Return to a boil, reduce to a simmer, and cook, partially covered and occasionally pushing chicken and squash into liquid, until chicken is cooked through and squash is tender, about 40 minutes. Season with more salt. Serve over rice, topped with yogurt and cilantro.