Preheat the oven to 400°F (200°C).
In a bowl, toss the halved baby potatoes with 1 tablespoon olive oil, 1/2 teaspoon salt, and 1/4 teaspoon black pepper. Spread them out on a baking sheet in a single layer.
Roast the potatoes in the preheated oven for 20-25 minutes, turning once halfway through, until golden and tender.
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. Season both sides with 1 teaspoon 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 set aside on a plate, covering loosely with foil to 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, stirring frequently to prevent burning.
Add the lemon juice to the skillet and stir to combine, cooking for another 1-2 minutes until the sauce slightly thickens.
Remove the skillet from heat and stir in the chopped fresh parsley.
In a separate skillet, heat 1 tablespoon olive oil over medium heat. Add the trimmed green beans, season with 1/4 teaspoon salt and 1/4 teaspoon black pepper, and sauté for 5-7 minutes until tender but still crisp.
To serve, divide the roasted potatoes and sautéed green beans evenly among four plates. Place a white fish fillet on each plate and drizzle with the lemon butter sauce.