Storage Notes: Cooled fish can be stored in an airtight container for up to 2 days.
Position the rack in the middle of the oven and preheat to 400 degrees.
Place the fish fillets in a medium baking dish. Use 1 tablespoon of the oil to brush both sides of the fish, then sprinkle with 1/8 teaspoon each salt and pepper. Place in the oven and roast for about 10 minutes per inch thickness, until the fish flakes easily with a fork. Remove the fish from the oven and cover with foil to keep warm.
While the oven preheats and the fish cooks, make the vegetables and broth. In a large, deep skillet over medium heat, heat the remaining 2 tablespoons of oil until shimmering. Add the garlic and cook until it turns pale golden brown, 1 to 2 minutes. Add the wine, increase the heat to high and cook until it is reduced by about half, 1 to 2 minutes. Add the chicken broth and the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Bring to a boil, then add the potato slices.
Reduce the heat to medium and simmer, uncovered, stirring occasionally, until the potatoes are nearly tender, about 10 minutes. Add the green beans and cook until they are crisp tender and the potatoes are tender, about 4 minutes more. Stir in the pesto and the lemon juice.
To serve, place a fish fillet onto each of the 4 large, shallow serving bowls or rimmed plates. Using a slotted spoon, distribute the vegetables among the plates, then spoon the broth over each.