Position a rack in the center of the oven; preheat to 425 degrees . In a medium bowl, stir together the two vinegars, brown sugar, worcestershire, soy sauce, mustard and chopped rosemary; season with salt and pepper. Whisk in 9 tbsp. EVOO.
In a large pot, bring 2 qt. water and 1 tbsp. salt to a boil. Fill a medium bowl halfway with ice water. Add the string beans to the boiling water and cook until brightly colored and al dente, about 1 minute; drain and plunge immediately into the ice bath to cool completely. Drain again, then pat dry.
Lay the salmon fillets skin side down on a greased baking sheet. Tuck a rosemary sprig under each. Brush the fillets with EVOO; season with salt and pepper. Whisk the sauce and spoon some over the fillets. Roast until firm and just cooked through, about 5 minutes. Remove and tent with foil.
In a medium skillet, melt the butter over medium heat; cook until light golden, 1 to 2 minutes. Immediately add the toasted pine nuts and cook for another 30 seconds, stirring to coat with the butter. Add the beans, season with salt and pepper and cook, stirring occasionally, until hot, about 2 minutes. Divide the beans and pine nuts between 2 plates.
Using a spatula, transfer the salmon fillets to the plates and drizzle each with another spoonful of sauce. Serve immediately.