Bring a large pot of water to a boil, then add a small handful of salt. Add the pasta and cook according to the package directions until al dente. Drain the pasta and reserve the pot.
While the pasta is cooking, melt 1 tablespoon of the butter in 2 tablespoons of the oil in a large skillet. Season the salmon with salt and pepper, then dust with flour. Add the salmon to the skillet and cook over medium-high heat, flipping occasionally, until golden on the outside and just cooked through, about 8 minutes.
In the reserved pot, melt 3 tablespoons of the butter in the remaining 4 tablespoons olive oil over low heat. Return the pasta to the pot, add all but 2 tablespoons of the herbs and toss well to coat. Season with salt and pepper.
Transfer the salmon to a plate; tent with foil to keep warm. Add the chicken broth, lemon juice and capers to the skillet. Simmer over high heat, scraping up any browned bits from the bottom of the skillet, until the liquid is slightly thickened, 3 to 5 minutes. Remove from the heat and gradually swirl in the remaining 2 tablespoons butter until emulsified. Stir in the reserved 2 tablespoons herbs and season with salt and pepper.
Add the salmon to the sauce and turn to coat. Divide the pasta among plates, top with the salmon and sauce and serve right away.