Preheat oven to 400° F.
Season chicken as desired and cook on an indoor or outdoor grill; cover to keep warm.
Chop broccoli into bite-size pieces and add to a large zip-lock bag, along with the oil. Seal well and shake to evenly coat.
Add broccoli to a baking dish and roast for 15-20 minutes until soft and starting to brown.
Meanwhile, cook pasta according to the package directions.
For the pesto: add 1/4 cup oil, juice from the lemon, minced garlic, basil, and walnuts to a food processor and blend until smooth.
Serve pasta topped with pesto, broccoli, and sliced chicken; serve grapes on the side.