Place the salmon fillets on a paper towel-lined plate. Pat each dry and season with the salt and pepper. Fill a large pot with water and add about 1 Tablespoon of salt. Bring to a boil while you sear the salmon.
Heat a large skillet over medium-high heat. Add the olive oil. Place the fillets skin side down. Sear for about 5-6 minutes without touching. Gently loosen the skin from the skillet and flip. Sear for another 4-5 minutes or until cooked through to your liking. The internal temperature should be at least 140 degrees.
Remove the salmon from the pan. Set aside on a clean plate and tent with foil. Reduce to medium heat. Add the shallots to the skillet that the salmon was seared in. Do not wipe it out. Sauté the shallot for 4-5 minutes.
While the shallots sauté and the wine reduces, add your pasta to the large pot of salted water and boil for one minute less than the shortest time recommended on the package. Once the pasta is about 2-3 minutes from being done, stir in the green beans. Drain and reserve about 1 cup of pasta water.
Add the garlic and sauté for another 1-2 minutes.
Deglaze the pan with the wine, scraping any brown bits off the bottom of the skillet.
Reduce the wine for 5 minutes.
Add about ½ cup of the reserved pasta water and stir until combined.
Stir in the cold butter until melted. Reduce to medium low heat.
Add the cooked pasta, green beans, lemon juice, zest, and parmesan cheese. Toss until well combined. Cook for 1-2 minutes.
Turn the heat off and stir in the pesto. Add more pasta water if needed to loosen the sauce.
Serve the salmon skin side up over the pasta with extra lemon wedges.