While the linguine is cooking, heat oil in a large skillet over medium-high heat. Add onion and garlic and cook 3 minutes, until soft. Add thyme and cook 1 minute, until fragrant. Add vermouth and cook 1 minute. Add broth, olives, lemon juice, capers, and lemon zest and bring to a simmer, for 5 minutes.
When the linguine is cooked, drain and add, along with the salmon, to the pan with the olive-caper mixture. Toss to combine using tongs and cook 1 minute to heat through. Season, to taste, with salt and freshly ground black pepper. Remove from heat and stir in basil. Serve immediately.