Wildly nutritious broccoli gets a starring role in this zingy, substantial pasta dish. For the highest yum factor, try to get your hands on a bunch of super-freshly grown broccoli -- and don't discard the stems! Peeled and sliced, they add a sweet, mild crunch to this lightly sauced and deeply flavorful pasta.
Bring a large pot of pasta-cooking water to a boil and salt it heavily: it should taste as salty as seawater. Cook the pasta (if you add it when the chicken is about half cooked, the sauce and pasta will be done at the same time), drain it, and return it to the pot with 3 tablespoons of the butter.
Pat the chicken dry and season on both sides with salt and pepper. In a very large skillet, heat the olive oil and remaining 1 tablespoon of butter over medium-high heat until foamy and very hot, then add the chicken and sauté until well browned and cooked through, around 4 minutes per side (thicker breasts may take longer; turn the heat down as needed to keep the chicken from burning). Remove the chicken to a cutting board.
Turn the heat to low and add the garlic to the pan, stirring for a few seconds until it is fragrant but not browned, then add the broccoli, the broth, and 1 teaspoon of salt, stirring to dissolve the browned bits on the bottom of the pan.
Cover the pan, turn the heat up to medium, and cook/steam until the broccoli is just tender: 3-4 minutes.
Add the chicken, which you've sliced thinly, back to the pan, along with the lemon juice and zest, the walnuts, and the buttered pasta, and stir to combine well (if the skillet isn't large enough, combine in the pasta pot).
Now taste a piece of pasta: it should be lemony and flavorful; add more salt, pepper, and lemon juice to taste. Stir in the parmesan cheese, then serve, passing more parmesan at the table.