Preheat oven to 350°. Cut squash in half down the middle, from top to bottom. Scrap out pulp and seeds. Place on a sheet pan, cut side down and bake for about 60 minutes, or until very tender.
Meanwhile, heat 1 tbsp sesame or canola oil in a Dutch oven over medium heat. Pat chicken thighs dry with a paper towel and season on both sides with a sprinkle of salt and pepper. Place skin side down in the pot and cook for 5-7 minutes, or until skin is golden brown. Remove to a plate.
Without wiping out the pan, add onions and mushrooms. Cook, stirring occasionally, until onions have softened and mushrooms have shrunken and are starting to brown.
Add garlic, carrots, and potatoes. Mix to combine.
Add rice wine vinegar, coconut milk, curry powder, coriander, cumin, and 1 tsp salt. Mix together and let it come to a simmer.
Place chicken thighs skin side up on top of the stew. Cover, remove from heat, and transfer to the oven. Cook for 40 minutes or until vegetables are tender and chicken is cooked through.
While stew is cooking, begin making rice: in a medium saucepan combine rice with chicken stock or water. Bring to a boil over high heat, then cover and reduce heat to low. Simmer for 15 minutes or until no liquid remains. Keep covered until ready to serve.
When squash is done roasting remove from oven and allow to cool until able to handle. Use a paring knife to cut the skin and peel it away from the flesh. Place the flesh in a bowl and mash until fairly smooth. Set aside.
When stew is done cooking, remove chicken thighs to a cutting board. Allow to cool slightly then remove skin and bones and cut chicken into cubes. Mix into stew along with 1 cup of mashed squash.
Serve stew with rice. Top with cilantro.