Bring a large pot of salted water to the boil and cook pasta according to package directions. Four minutes before the pasta is done, add broccoli florets.
In the meantime, season chicken with salt and pepper. 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, 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.
Drain pasta, reserving 1/4 cup of the pasta cooking water.
Add pasta, reserved pasta cooking water, Parmigiano-Reggiano cheese and chicken then toss. Garnish with green onions.