Cut the squash in half, length wise. Cook the squash by placing both halves in boiling water until fork tender.
While squash is cooking, in a large bowl, mix sweet white rice flour and salt with 1/3 cup hot water, until dough forms.
Knead dough into a ball and wrap with plastic wrap. Set aside for 30 minutes.Drain cooked squash. Spoon out the flesh and place into bowl.
Add spices and brown sugar.Add squash mixture, milk, and 3 cups of water to stock pot. Bring to a boil and take off heat.
Using hand mixer, blend the squash mixture to desired consistency. Place back on heat and simmer.
While stock is simmering, knead mochi ball dough. Pull off a piece of dough and roll into a ball about the size of a gumball (a little smaller).
Add mochi balls to soup.While soup is cooking, sauté mushrooms in sesame oil.Remove soup from heat once mochi balls have risen to the top. Stir in sautéed mushrooms.Serve with sautéed mushrooms on top as garnish.