Heat vegetable oil in a large stockpot or Dutch oven over medium heat. Season the chicken pieces generously with salt and pepper. Add chicken to the pot and sear for about 2 to 3 minutes until lightly browned on both sides, then transfer to a plate and set aside.

Lower the heat to medium-low and melt 1 tablespoon of butter in the same pot. Add the onions and carrots, then sauté for 3 to 4 minutes, stirring occasionally, until the onions soften. Add the shimeji mushrooms and cook for another 2 to 3 minutes until fragrant.

Add the remaining 1 tablespoon of butter to the pot. Once melted, sprinkle the flour evenly over the vegetables and stir continuously for 1 to 2 minutes to cook out the raw flour taste.

Pour in the white wine, using a wooden spoon to scrape up all the browned bits from the bottom of the pot. Return the seared chicken and its juices back to the pot, then add the water and vegetable bouillon. Bring to a boil, reduce heat to low, cover with a lid, and simmer for 15 to 20 minutes.

Stir the kabocha squash pieces into the stew. Cover again and simmer for another 10 to 15 minutes, or until the kabocha is fork-tender.

Turn the heat to low and stir in the heavy cream. Cook gently for 2 to 3 minutes until thoroughly heated through and thickened. Taste and adjust with a final seasoning of salt and pepper before serving.