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Using a vegetable peeler, peel 3 (about 3 inch long) strips of lemon peel from 1 of the lemons and add to a mason jar or airtight container. Reserve peeled lemon.
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Add olives, olive brine, gin, and vermouth to jar with lemon peels. Seal and shake well to combine. Chill in refrigerator until flavors meld, at least 30 minutes and up to 24 hours. Strain marinated olives using a fine-mesh sieve set over a medium bowl, reserving olives and brine mixture separately; discard lemon peel strips.
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Preheat oven to 275°F. Remove salmon from refrigerator 15 minutes before cooking. Thinly slice peeled lemon and 1 of the remaining lemons and place in the bottom of a 13- x 9-inch baking dish. Place salmon, skin-side down, on top of lemon slices in baking dish and season evenly with pepper and 1/2 teaspoon salt. Stir together 2 tablespoons olive oil and 2 tablespoons of the reserved brine mixture; spoon over salmon.
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Bake until cooked to desired degree of doneness, 30 to 35 minutes for medium-rare (about 120°F) and 35 to 40 minutes for medium (about 125°F).
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Meanwhile, squeeze 1 tablespoon lemon juice from remaining lemon into a medium bowl. Stir in parsley, chives, basil, garlic, crushed red pepper, reserved olives, 1 tablespoon of the reserved brine mixture, and remaining 6 tablespoons oil and 3/4 teaspoon salt until combined. Top salmon with olive mixture and serve with lemon wedges.