Preheat the oven to 250 degrees F.
Place the salad greens in a large bowl. Zest the lemon over the salad greens; squeeze in the juice of half of the lemon (reserve the other half). Drizzle the greens with 2 tablespoons of the olive oil, and season with salt and pepper to taste; toss. Set aside.
Sprinkle the salmon fillets all over with salt and several grinds of black pepper. Heat a large nonstick skillet over medium-high heat. When the skillet is hot, add the remaining 1 tablespoon olive oil. Add the fillets and cook until well browned, 2 to 3 minutes. Turn the fillets and cook until just cooked through, another 2 to 4 minutes depending on thickness. Transfer the salmon to a baking sheet and keep warm in the oven while you make the sauce.
Return the skillet to medium heat and add the shallots and capers. Cook, stirring occasionally, until the shallots have softened, 2 to 3 minutes. Add the rose and the juice of the reserved lemon half. Bring to a boil, lower the heat and simmer until reduced by half, about 2 minutes. Whisk in the butter, a few cubes at a time, to make a creamy sauce. Stir in the parsley and chives.
Serve the sauce over the salmon, with the dressed greens alongside.