Place shiitake mushrooms in a medium bowl and cover with 2 1/4 cups water.
Allow to soak 1 to 3 hours.
Remove mushrooms, reserving the liquid, and squeeze mushrooms over the bowl.
Remove and discard mushroom stems, and slice mushrooms into 1/4 inch strips.
In a medium-size (about 4 quarts) clay pot (or other kind of pot) over medium heat, combine 1 1/4 cups reserved mushroom liquid, sake, mirin and soy sauce.
Add rice and stir gently.
Top rice with mushrooms, burdock, hijiki, carrots and edamame; do not stir.
Cover, and raise heat to bring to a boil.
Reduce heat to low and simmer for 16 minutes.
Turn off the heat and leave covered for 8 minutes.
Using a spatula, fluff up the rice, and garnish with a sprinkling of sesame seeds.
If desired, serve accompanied by Japanese pickled silced daikon, hijiki, or cucumbers, or a green salad.