In a small saucepan, simmer the mirin over high heat, whisking constantly, for 3-5 minutes until the alcohol smell disappears. Whisk in the sugar until dissolved and then the soy sauce. Remove from heat.
Mix 6 tablespoons of this mixture into the dashi. Set aside the remaining and store in the refrigerator for up to a year.
To assemble the soba:
Bring a large pot of water to a boil. Add the soba and cook until just tender, 4-5 minutes. Drain and place in a large bowl filled with ice water. Swirl the noodles, drain, rinse, and divide among serving plates.
Grind the walnuts in a small food processor or mortar and pestle until just powdery.
Place the powdered walnuts and thinly sliced green onions in small bowls next to the mori tsuyu (dipping sauce). Season the dipping sauce with desired amounts of walnuts and green onions. Dip each bite of soba noodles into the sauce before eating.