Cut salmon fillet into two equal portions and brush both sides of each piece with olive oil. Sprinkle flesh side with salt, pepper and dried thyme. In a large skillet, heat one tablespoon olive oil and butter on medium-high until butter foams. Carefully place fillets flesh-side down in skillet. Cook without moving until edges begin to crisp, about 2 minutes. Decrease heat to medium and cook until fillets are golden brown, about 1 minute more. Turn fillets so they are skin-side down and continue cooking until medium-rare and skin is crisped, 5-7 minutes depending on their thickness. Add a few additional drops of olive oil as needed to prevent sticking. Remove fillets to serving plates and drizzle with balsamic vinegar. Serve immediately.
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