In a small bowl, whisk together 3 tablespoons soy sauce, 2 tablespoons rice wine vinegar, 2 teaspoons sesame oil, 2 cloves grated garlic, and the grated ginger.
Add steak to a sealable bag and pour half of the marinade into the bag (about 2 tablespoons). Seal the bag and let the steak marinate for at least 30 minutes or up to 8 hours.
In the meantime, make the eggs. Fill a saucepan with 3 inches of water and bring to a boil over medium heat.
Gently lower the eggs into the boiling water. Boil for 6 minutes.
Remove the eggs from the water and plunge immediately into a bowl of cold water and ice.
Remove the shells from the eggs and slice in half.
Heat the oil in a skillet over medium heat.
Add the mushrooms and cook until glossy, about 5 minutes.
Add 1 tablespoon of the reserved steak marinade to the mushrooms and toss to coat. Remove from the skillet and set aside.
Add the broccolini and cook until bright green and tender, about 5 minutes.
Add the remaining marinade and stir to coat, then remove and reserve the broccolini.
In a small bowl, whisk together the miso, 1 tablespoon soy sauce, 1 tablespoon rice wine vinegar, 1 tablespoon chili sauce, 1 teaspoon sesame oil, 1 clove grated garlic, and the honey.
Add the udon noodles to the skillet with ½ cup water. Gently toss over medium heat until noodles separate.
Add the miso sauce and toss with the udon noodles.
Remove the udon noodles and divide among bowls.
Add the steak to the skillet, adding more oil if needed. Sear on either side until browned, about 2 minutes per side.
Remove the steak from the skillet and slice.
Add the vegetables, eggs, and steak to the noodles in the bowls to serve. Optionally garnish with sesame seeds.