Place butter, brown sugar, garlic, lemon juice, soy sauce, and pepper in a small saucepan. Warm over medium heat, stirring occasionally, until sugar is completely dissolved.
Lightly oil grill grate. Brush fish with brown sugar sauce, and place on grill. Cook for 5 minutes per side, or until fish can be easily flaked with a fork, basting with sauce. Discard remaining basting sauce.