Slice chicken into 1/4 inch thick slices and toss with 2 tbsp soy sauce and 1 tbsp mirin. Sprinkle cornstarch over it and toss to coat
Place whites and pale greens of scallions in a food processor with garlic and ginger in a food processor until finely chopped (reserve dark green part of scallions)
Heat 2 tbsp vegetable oil in a large skillet or wok over medium-high. Arrange chicken in skillet and cook, undisturbed, until golden brown underneath (about 3 minutes). Turn chicken over and cook until golden brown on the other side, about 2-3 more minutes. Transfer to plate
Heat remaining 2 tbsp vegetable oil in same skillet or wok. Add scallion/garlic/ginger mixture and cook, stirring, for about 30 seconds. Add bok choy (sliced in half) and snow peas and cook until crisp-tender, tossing often
Return chicken and any accumulated juices to the skillet or wok. Add remaining 3 tbsp soy sauce and 1 tbsp mirin as well as 1/4 cup water. Scrape up any browned bits stuck to bottom of pan. Cook, tossing, until vegetables are tender and sauce is slightly thickened, about two minutes. Remove from heat; season with salt and pepper as necessary
Divide amongst bowls and serve topped with dark green part of scallions, thinly sliced with cooked rice alongside