Combine the flaked salmon, green onion, mayonnaise and pinch of salt and pepper. Stir until well combined. Set aside.
Stir the soy sauce into the rice. Line a bowl with plastic wrap, and spoon some of the rice into the bowl. Spoon the salmon mix into the middle of the rice. Use another spoonful of rice to cover the salmon. Firmly pack the rice until you have a ball. Use the plastic wrap to form the rice into a ball. Set aside while you work on the details.
Hole-punch two eyes and a nose out of the nori. Cut a circle out of the cheese for the face. Slice rounds of hot dog for the ears. Cut three circles out of the slice of white cheese, two smaller circles for the brown bear's ears and a larger one for its face. Stick the nori, cheese and hot dog onto your brown bear rice ball.