Place a large pot of salted water over high heat to boil. Once at a boil, add the pasta and cook to al dente according to package directions. Drain cooked pasta, reserving a large mug of the starchy cooking water, and reserve in colander.
While pasta is cooking, place a medium skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Season the chicken with salt and pepper, and add it to the hot pan. Cook the chicken, stirring occasionally, until golden brown and cooked through, 6-7 minutes.
While the chicken is cooking, place a small saucepan with about an inch of water to a boil over medium-high heat. Add the broccoli and cook until tender, about 1 minute. Remove broccoli from water and add to pan with chicken, tossing to combine.
When pasta has finished cooking, return pasta pot to the stovetop over medium heat and add the butter to melt. Add the Parmigiano-Reggiano and toss the pasta in the melted butter and cheese to coat.
To serve, divide the pasta between four plates, twirling it into a nest shape on the plate as you portion it. Divide the chicken and broccoli between the four plates, placing it inside the pasta "nest."