1. With a very sharp knife, make shallow slashes about 1 inch apart in skin of the snapper fillets. Season the fillets on both sides with 1/2 teaspoon salt and 1/4 teaspoon pepper. Heat the olive oil in a large, nonstick frying pan over medium-high heat. Add the fish, skin side down, and cook for 2 1/2–3 minutes. Gently press the fillets down with a spatula to flatten while cooking. Turn fillets, and cook an additional 1–2 minutes or until cooked through. Remove from heat, and keep warm.
2. Whisk together the vinegar, honey, Dijon, and chives in a small bowl. Slowly drizzle in the extra-virgin olive oil and water, whisking constantly. Season the vinaigrette with the remaining 1/8 teaspoon salt and several turns freshly ground black pepper.
3. Combine arugula and blueberries in a large bowl. Drizzle vinaigrette over salad, tossing until well combined. Divide salad evenly among 4 plates. Arrange 4 pecorino slices over each salad. Serve immediately with seared snapper and a 5-ounce glass of white wine, if desired.