Combine water and dried kelp in a pot. Heat over medium heat until the water is almost boiling - bubbles beginning to appear on the sides of the pot. Remove the dried kelp and set aside.
Add the dried bonito flakes into the pot and give it a gentle stir to make sure all the bonito flakes are submerged in water. Bring to a boil over medium heat, then turn off the heat. Let the bonito flakes sit for 5 minutes.
Place a sieve over a big bowl and line a cheesecloth on top. Strain the stock over the cheesecloth. Pick up the edges of the cheesecloth, twist it and squeeze the bonito flakes to extract every last bit of goodness (be careful as it will be very hot!).
Niban Dashi
Add the set aside kelp and the bonito flakes in the cheesecloth back into the pot. Add in water and shiitake mushrooms (if using).
Cover the pot with a lid then bring to a boil over high heat. Turn the heat down to a simmer and simmer for 10 minutes. Turn off the heat heat and set aside for another 10 minutes.
Discard the kombu. Remove the shiitake mushrooms and use if for other recipes (like Chawanmushi). Strain the stock over the cheesecloth. Likewise, pick up the edges of the cheesecloth, twist it and squeeze the bonito flakes to extract every last bit of goodness.
NOTES
After I use the Ichiban Dashi (usually for chawanmushi), I will combine the remaining Ichiban Dashi (in which there won't be a lot left) with the Niban Dashi and freeze them in small containers (in ice cube trays or in 1-cup quantities).
Alternatively, I will freeze the strained bonito and kombu after making the Ichiban Dashi and make a big batch of Niban Dashi the next time I require dashi stock.