Combine oyster sauce, soy sauce, mirin, rice vinegar, Worcestershire sauce, sesame oil, sugar, Sriracha, and garlic in a small bowl; set aside. Trim and discard root ends of scallions. Chop whites and light greens into 2-inch pieces and quarter lengthwise; set aside. Thinly slice dark green parts of scallions and keep for garnish.
In a large skillet or wok over medium-high heat, heat the oil. Add the mushrooms in a single layer and cook without stirring until the first side is browned, approximately 3 minutes. Stir and continue cooking, stirring occasionally, until the mushrooms are tender and golden brown on both sides, about 4 more minutes. Add bok choy, carrots, sliced white and light green parts of scallions, and 1/4 cup of water to the pan with the mushrooms. Cook, stirring occasionally, until the vegetables are just tender, around 3 minutes.
Add the udon noodles and the remaining 1/4 cup of water, gently separate the noodles with tongs or a spoon. Pour in the reserved oyster sauce mixture and stir constantly until the noodles and vegetables are well-coated, which should take about 1 minute. Transfer to serving bowls, top with the reserved sliced scallion greens and any optional garnishes you like. Serve right away.