Cook bok choy in pot of boiling salted water until crisp-tender, about 2 minutes. Drain. Set aside. Heat 2 teaspoons vegetable oil in heavy large skillet over medium heat. Add curry paste and stir 30 seconds. Add coconut milk, chopped lemongrass and 1 tablespoon brown sugar. Boil until mixture is slightly thickened and reduced to 1 cup, about 5 minutes. Stir in 2 1/2 tablespoons lime juice and 1 tablespoon fish sauce. Set curry sauce aside.
Preheat oven to 400°F. Heat 1 tablespoon oil in heavy large ovenproof skillet over high heat. Sprinkle salmon with salt and pepper; add to skillet. Cook salmon 2 minutes per side. Transfer skillet to oven. Bake salmon until opaque in center, about 6 minutes. Meanwhile, heat remaining 1 tablespoon vegetable oil in large nonstick skillet over medium-high heat. Add bok choy and sauté until beginning to brown in spots, about 4 minutes. Drizzle bok choy with remaining 1 tablespoon fish sauce. Bring curry sauce to simmer and ladle onto 4 plates.
Arrange 1 salmon fillet and 2 pieces of bok choy atop curry sauce on each plate and serve.