Make the dressing: In a food processor or blender, process 1 cup of the sesame seeds with the sesame oil at high speed until smooth. Add the rice vinegar, soy sauce and ginger and pulse to combine. Transfer to a bowl and whisk in 2 tablespoons of the dashi. Set aside.
Bring a large pot of salted water to a boil. Blanch the snow peas until crisp-tender, about 2 minutes. Transfer to paper towels, blot dry and let cool completely.
In a medium skillet, heat the remaining ½ cup dashi over medium heat. Add the mushrooms and cook, stirring occasionally, for 3 to 4 minutes until the mushrooms are tender and most of the liquid is absorbed. Set aside to cool completely.
In a large bowl, combine the snow peas, mushrooms and tofu. Add ¾ cup of the sesame dressing and toss to coat. Divide among four bowls and garnish with the remaining 2 tablespoons of sesame seeds. Serve the extra dressing on the side.
