Season the Ahi with salt and pepper. Heat 2 tablespoons of olive oil in a saute pan over medium-high heat. Sear the Ahi quickly until medium rare, or to your liking, about one minute on each side.
Set the Ahi aside and cook mushrooms in the same pan. Deglaze with sherry and reduce to medium heat. Add the scallions, teriyaki glaze, and buerre blanc to the pan and heat until combined and slightly thickened, about 3 minutes.
In another saute pan, heat the remaining olive oil and add the diced vegetables. Cook for 1 minute. Add the wild and basmati rice and season with salt and pepper to taste. When the vegetables are cooked and the rice is warm, pour the Japanese vinaigrette over the rice mixture.
Place 5 ounces of the rice mixture in the center of each plate. Pour two to three ounces of the Teriyaki buerre blanc mixture on 1/2 of the plate. Slice a 5-ounce portion of the Ahi into strips and fan over the rice and sauce. Garnish with scallions.
Combine the soy sauce, brown and white sugars, garlic, ginger, Japanese vinegar, and sake, and in a heavy-bottomed saucepot. Bring to a boil (the sauce will flame as the alcohol burns off). Dissolve the arrowroot in the water. Reduce the sauce to a hard simmer and add the arrowroot mixture. Simmer hard for 5 minutes more.
Beurre Blanc: Combine the wine, shallots, and vinegar in a small saucepan. Bring to a boil over medium-high heat. Continue boiling until 1 tablespoon of liquid remains. Lower the heat and whisk in the butter, piece by piece. When the butter is completely incorporated, remove from heat and salt to taste.
Japanese Vinaigrette: Combine first 5 ingredients in a stainless steel mixing bowl until blended. Using a wire whisk, pour in sugar and emulsify.