Bring a large pot of lightly salted water to a boil. Add Penne pasta and boil according to pkg directions. Add the broccoli to the pasta for last 3 minutes of cooking. Drain and set aside. Lightly season chicken breasts on each side with salt and pepper. Saute chicken in skillet with 1 Tbsp olive oil until cooked through, about 4-6 minutes per side, depending on thickness of chicken. Remove chicken to a plate and let it rest 5 minutes before slicing into strips.

In a 12-inch skillet, melt 1 Tbsp butter over medium heat and add onion and garlic. Saute, stirring often, 2-3 minutes, until onion is soft. Stir in chicken broth and simmer until it has reduced to about 1 1/2 cups, 6-12 minutes. Reduce heat to low and add the remaining butter, a few cubes at a time, swirling to melt before adding more. Stir in lemon juice, chives, and parsley. Season the sauce to taste with salt and pepper.

Transfer pasta and broccoli to a large bowl. Add tomatoes, artichokes, and olives. Pour sauce over top and toss to coat. Serve with slices of grilled chicken and shaved Parmigiano-Reggiano cheese.