Put the tofu cakes into a small pot and cover with water (about 3 cups). Bring to a boil. Lower the heat and simmer very gently for 10 minutes. Drain.
Meanwhile, prepare the vegetables. French cut the beans into 5 centimeter / 2 inch long diagonal strips. Trim, peel and cut the carrots into very fine 5 centimeter / 2 inch julienne strips.
Put the drained bean curd, the roasted and ground sesame seeds, sugar, mirin, salt, soy sauce and dashi into the container of a food processor or blender. Blend until you have a paste. Set aside.
Bring two pots of water to a boil. When boiling, drop the 'frenched' beans into one pot and the carrots into the other. Drain the carrots after 2-3 minutes. They should be cooked through but still crunchy. Drain the beans after about 4 minutes. They should be cooked through but still have a bite to them.
Put the carrots and beans in a bowl, add the dressing and mix well. Serve hot, warm or at room temperature.