Combine chicken or steak with 2/3 cup teriyaki sauce and marinate for 3 hours in the refrigerator. If using flank steak, shallowly score one side with a knife to allow a bit more flavor to penetrate the meat.
Remove meat from the marinade and heat a grill pan or skillet with a medium-high flame. Once heated, add either the chicken or beef and grill/cook until cooked through and the internal temperature reaches 165 degrees (for the chicken) or to desired level of doneness (for the beef). Once done, remove from the heat to a plate and then tent with foil. Set aside.
Heat the oil over medium-high heat in a large skillet until the oil is shimmering. Add the carrots and broccoli and stir-fry for about 2-3 minutes. Add the celery and onion and stir-fry for another 2 minutes. Next, add the water chestnuts, garlic and ginger. Stir-fry until the veggies are crisp tender.
Meanwhile, put on a large pot of water to boil and cook the udon noodles according to the package’s instructions (al dente). Reserve 1/2 cup of the cooking water and then drain and rinse the udon under cold water to remove gumminess.
Add the noodles to the hot skillet along with the remaining 2/3 cup teriyaki sauce and 1/2 cup of the reserved pasta water. Reduce heat to medium and start tossing the udon, vegetables and sauce together.
Simmer for a minute. If you’re serving flank steak, take this moment to slice it thinly. You can either add the flank steak/chicken directly to the pan and toss or else dish up the udon and top with grilled steak or chicken. Serve in bowls and garnish with green onions.