Pat salmon dry with paper towels and sprinkle with salt and pepper on both sides. Heat 1 tablespoon of oil in a large pan over medium heat. Sear salmon (skin side first) on each side until golden brown, about 4 minutes. Set cooked salmon aside on a clean plate tented with foil.
Add the remaining tablespoon of oil to the pan, along with the shallots, and cook until softened, about 4 minutes. Add wine and scrape up any bits from the bottom of the pan. Add chicken broth, lemon juice, and lemon slices and bring sauce to a simmer. Cook until the sauce thickens enough to coat the back of a spoon. Once thick, remove from heat, and add the butter, capers, and parsley. Taste and adjust seasoning to your liking.
Place 1 salmon filet on each plate and spoon sauce on top. Sprinkle with additional parsley, and enjoy!