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.