If you can’t find ripe mangoes, substitute thawed frozen mango chunks.
To make Mango-Teriyaki Sauce: Purée mango and ginger in food processor until smooth. Strain 1/2 cup purée into saucepan; reserve remaining purée for another use. Whisk in mirin, soy sauce, sake, sugar, and 2 Tbs. water. Bring to a simmer over medium heat, stirring constantly. Remove from heat.
Combine rice, salt, and 2 cups water in saucepan. Bring to a simmer over medium-high heat. Reduce heat to low, cover, and cook 25 minutes. Let stand, covered, 15 minutes.
Coat tofu slices with Mango-Teriyaki Sauce. Heat grill or grill pan over medium-high heat. Spray with cooking spray. Grill tofu 3 minutes. Flip, and brush with more Mango-Teriyaki Sauce. Grill 2 minutes more.
Stir 1/3 cup green onions, sesame seeds, and sesame oil into rice. Mound rice onto plates. Arrange tofu atop rice. Drizzle with remaining sauce. Sprinkle with remaining green onions, and serve.