Toss 1½ lb. skinless, boneless chicken breasts, sliced on a diagonal ¼" thick, with 2 Tbsp. soy sauce and 1 Tbsp. mirin in a medium bowl. Sprinkle 1 Tbsp. cornstarch over; toss to coat.

Pulse white and pale green parts of 4 scallions, 6 garlic cloves, coarsely chopped, and one 2" piece ginger, scrubbed, coarsely chopped, in a food processor until very finely chopped.

Heat 2 Tbsp. vegetable oil in a large skillet over medium-high. Arrange chicken in skillet in an even layer and cook, undisturbed, until golden brown underneath, about 3 minutes. Turn chicken over and cook until golden brown on other side, about 2 minutes. Transfer to a plate.

Heat remaining 2 Tbsp. vegetable oil in same skillet. Add scallion mixture and cook, stirring, until fragrant, about 30 seconds. Add 8 oz. baby bok choy, halved lengthwise, 8 oz. mushrooms, trimmed, sliced, and 8 oz. snow peas, trimmed, and cook, tossing, until crisp-tender, about 2 minutes. Return chicken and any accumulated juices to skillet. Add remaining 3 Tbsp. soy sauce and 1 Tbsp. mirin and ¼ cup water, scraping up any browned bits stuck to bottom of pan. Cook, tossing, until vegetables are tender and sauce is slightly thickened, about 2 minutes. Remove from heat; taste and season with kosher salt if needed.

Divide stir-fry among plates; top with dark green parts of 4 scallions. Serve with cooked rice alongside.