Bring a large pot of salted water to the boil and cook noodles according to package directions. Four minutes before the noodles are done, add broccoli florets.
In the meantime, season chicken with salt and pepper. Cut into 1-inch pieces. Add olive oil to a large skillet over medium-high heat, add chicken and cook until browned on all sides and cooked through, about 6 minutes.
Remove chicken and set aside. Turn heat to low then add butter to skillet. Once melted, remove pan from heat and add miso, whisk until butter and miso come together then add Greek yogurt, stir to combine. Add cooked chicken to mixture and toss well.
Drain noodles, reserving 1/4 cup of the pasta cooking water.
Place drained noodles in a large serving bowl. Add reserved noodle cooking water, chicken mixture, and Parmigiano-Reggiano cheese. Toss all ingredients well to mix. Garnish with green onions.
Cook the pasta according to package directions. Drain and set aside.
Heat a wok or large sauté pan over medium heat. When hot, add the butter, garlic and green onion. Cook for 1 minute or until very fragrant, making sure to keep an eye on the heat so that the garlic does not burn. Add in the mushrooms and cook for 2 minutes.
Remove from heat, add in the miso, oyster sauce and stir.