In a medium bowl, stir together the soy sauce, mirin, ginger and ¼ teaspoon pepper. Add the mushrooms and carrots, then toss to coat. In a small bowl, combine the butter and miso and stir until homogenous; set aside at room temperature.
In a medium saucepan, combine the rice and 1½ cups water. Bring to a simmer over medium-high, then reduce to low. Add the mushroom mixture and any liquid in the bowl, scattering it in an even layer on top of the rice; do not stir. Cover and cook until the rice has absorbed all of the liquid and the mushrooms and carrots are softened, about 20 minutes. Remove from the heat and let stand, covered, for 10 minutes.
Uncover and add the miso butter. Using a silicone spatula, fluff the rice, stirring to incorporate the vegetables and butter, until well combined. Taste and season with salt and pepper.
Optional garnish: Thinly sliced scallions
