Combine the rice with 1 1/4 cups cold water in a heavy-bottomed pot with a lid. Bring to a boil over high heat, stir once, then lower heat to low, cover, and cook for 20 to 22 minutes until the rice is completely tender and all the liquid is absorbed.

Empty the hot rice into a large glass or ceramic bowl. Sprinkle with 2 tablespoons rice vinegar, the sugar, and salt. Fold gently with a wooden spoon or rice paddle until the vinegar is absorbed. Taste. It should have a mild, seasoned vinegar flavor with a hint of sweetness. Cover with plastic wrap and set aside. Rice should still be slightly warm when you use it.

While the rice cooks, place the chopped yam in a small pot, cover with water, and bring to a boil. Once boiling, lower heat and cook for about 10 to 12 minutes until completely tender when pierced with a fork. Drain and transfer to a bowl. Mash with the sesame oil until smooth but still a little textured. Set aside.

Place the shiitake mushrooms in a small saucepan with the 1/2 cup water, soy sauce, and mirin. Bring to a simmer and cook for about 10 minutes until the mushrooms are tender and have absorbed most of the liquid. Let cool slightly then slice into strips.

In a small shallow cup, combine 1/3 cup water and 1 tablespoon rice vinegar. Keep near your workstation.

Place a nori sheet on the bamboo mat. Moisten your hands with the vinegar water, scoop a scant 1 cup of rice, pat it gently together, and spread it onto the bottom two thirds of the nori sheet in a layer less than a third of an inch thick, all the way to the edges. Spread about 2 tablespoons of mashed yam across the center of the rice and layer the mushroom strips on top. Sprinkle with sesame seeds.

Roll from the rice end, keeping a firm grip. When you reach the bare nori, pat with a little vinegar water to seal. Press gently to seal the whole roll. Slice into 1-inch pieces with a very sharp serrated knife.