Pat sablefish filets dry with paper towels and coat with ½ teaspoon salt and ½ teaspoon pepper.
Add olive oil to a large skillet over medium heat. Once hot, add sablefish filets skin-side down. Cook for 4 minutes and then flip. Cook for another 4-5 minutes, until the fish is starting to turn golden and flakes easily with a fork. If you're using a meat thermometer, you'll want the internal temperature to be 145°F.
While fish is cooking, melt butter in a small saucepan over medium heat.
Once melted, add in garlic and cook for 1 minute, until fragrant.
Stir in lemon juice and vegetable or chicken broth. Let mixture come to a simmer for 3-4 minutes until it thickens slightly. Mix in additional ¼ teaspoon salt and ¼ teaspoon pepper and taste. Add more if desired.
When ready to serve, spoon lemon butter sauce over sablefish filets and top with chopped parsley.