Preheat the oven to 400°F (200°C).
In a bowl, toss the halved baby potatoes with 1 tablespoon olive oil, 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper.
Spread the potatoes on a baking sheet in a single layer and roast in the preheated oven for 20 minutes, turning once halfway through cooking.
While the potatoes roast, heat 1 tablespoon olive oil in a large skillet over medium-high heat.
Pat the white fish fillets dry with paper towels and season both sides with 1 teaspoon kosher salt and 1/2 teaspoon black pepper.
Add the fish fillets to the hot skillet and cook for 3-4 minutes on each side, until the fish is opaque and flakes easily with a fork. Remove the fish from the skillet and keep warm.
In the same skillet, reduce heat to medium and add 2 tablespoons butter and the minced garlic. Cook for 1 minute until fragrant.
Add the lemon juice to the skillet and stir to combine, cooking for another 1-2 minutes to slightly thicken the sauce.
Remove the skillet from heat and stir in the chopped parsley.
In a separate skillet, heat 1 tablespoon olive oil over medium heat. Add the trimmed green beans, season with 1/4 teaspoon kosher salt and 1/8 teaspoon black pepper, and sauté for 5-7 minutes until tender-crisp.
Serve the white fish fillets topped with the lemon butter sauce alongside the roasted baby potatoes and sautéed green beans.