In a bowl, mix vinegar, brown sugar, and mustard.
Rinse halibut and pat dry. Brush lightly all over with oil and sprinkle with salt and pepper. Heat a 10- to 12-inch nonstick frying pan over medium-high heat. Add fish and cook until browned on the bottom, about 4 minutes.
Turn pieces over. Spoon vinegar mixture over browned sides. Lower heat and cook until fish is opaque but still moist-looking in the center (cut to test), 4 to 5 minutes longer. Transfer to plates.