Cut white and pale green parts of scallions crosswise into 2-inch pieces, then thinly slice enough scallion greens to measure 3 tablespoons.
Stir together brown sugar, soy sauce, and lemon juice in a small bowl until sugar is dissolved.
Pat fish dry and sprinkle with salt. Heat oil in a 12-inch nonstick skillet over moderately high heat until hot but not smoking, then cook fish on 1 side until browned, 3 to 4 minutes. Turn fish over and brown 1 minute. Add soy sauce mixture and simmer, covered, until fish is almost cooked through, about 3 minutes. Transfer fish with a slotted spatula to a heatproof platter and keep warm in oven. (Fish will cook through from residual heat.).
Add 2-inch scallion pieces and juices from fish platter to sauce and boil, stirring occasionally, until glaze is very thick and reduced to less than 1/4 cup, about 5 minutes. Spoon glaze over fish and sprinkle with scallion greens.