Gently stir 2 cups short-grain white rice, quinoa, mirin, 2 tsp. soy sauce, and 2½ cups water in a small Dutch oven or medium heavy saucepan with a tight-fitting lid to combine. Arrange shiitake mushrooms, thinly sliced, on top. Gently place all of the salmon fillets over mushrooms; lightly season fish with kosher salt.
Set pot over medium-high heat, cover, leaving lid slightly askew, and cook until small bubbles appear along the sides, about 5 minutes. Reduce heat to medium, cover pot tightly, and cook undisturbed 15 minutes. Remove from heat and let sit 20 minutes.
Meanwhile, mix rice vinegar, sesame oil, 2½ scallions, and remaining 3 Tbsp. soy sauce in a small bowl to combine. Set scallion-soy dressing aside.
Uncover rice mixture and using a rice paddle, carefully break apart salmon fillets into smaller pieces and fold into rice mixture. Mix in remaining 2½ scallions, thinly sliced. Transfer rice mixture to shallow bowls and serve with reserved scallion-soy dressing alongside for spooning over.