Bring a large pot of salted water to a boil. Add the spaghetti and cook as the label directs for al dente, adding the broccoli in the last 2 minutes of cooking. Reserve 1/2 cup cooking water, then drain the pasta and broccoli and return to the pot. Add 1 tablespoon butter, 2 tablespoons parsley, the lemon zest and a big pinch each of salt and pepper. Stir well, adding the reserved cooking water as needed to loosen.
Meanwhile, season the pork chops on both sides with salt and pepper. Dust with the flour, shaking off any excess. Heat the olive oil in a large skillet over medium-high heat. Add the pork chops and cook until well browned and just cooked through, about 4 minutes per side. Remove to a plate and tent with foil to keep warm.
Carefully add the wine to the skillet, scraping up the browned bits from the pan. Cook until reduced by about half, then stir in the lemon juice and capers; return to a simmer. Remove from the heat and swirl in the remaining 2 tablespoons each butter and parsley until the butter is melted. Pour in any accumulated juices from the plate of pork. Season the sauce with salt and pepper.
Divide the spaghetti and broccoli among plates and top each serving with a pork chop. Spoon the sauce on top.