Preheat oven to 350°.
Heat 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 pinch of salt and pepper. Place skin 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 onion and garlic and cook briefly until fragrant.
Add shiitake mushrooms and cook, stirring occasionally, until shrunken and starting to brown, about 5 minutes.
Add squash, potatoes, and flour and cook, stirring occasionally, for 1-2 minutes.
Add rice wine vinegar, coconut milk, curry powder, 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 very tender.
When stew is about halfway done, begin cooking 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. Cook for 15 minutes or until no liquid remains. Keep covered until ready to serve.
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 back into stew.
Fluff rice with a fork and divide into serving bowls. Ladle stew over rice.