Whisk together soy sauce, oyster sauce, mirin, brown sugar, and sesame oil in a small bowl until well combined and sugar dissolves. Set aside.
Pat shrimp dry using paper towels. Heat 1 tablespoon neutral oil in a wok or large cast-iron skillet over medium-high until shimmering. Add shrimp; cook, stirring often, until pink and just cooked through, 2 to 3 minutes. Transfer shrimp to a bowl.
Add remaining 2 tablespoons neutral oil to skillet. Add scallions and mushrooms to skillet; cook over medium-high, stirring often, until just starting to brown, 2 to 3 minutes, adjusting heat as needed to prevent overbrowning. Add cabbage, carrot, and garlic; cook, stirring constantly, until vegetables are starting to soften, 3 to 4 minutes.
Add udon, soy sauce mixture, and shrimp to skillet, stirring to coat. Cook, stirring constantly, over medium-high until noodles are tender and sauce is absorbed, about 2 minutes.
Garnished with bonito flakes (if using) and thinly sliced scallions; serve immediately.
