Preheat the oven to 400°F (200°C). Line a large sheet pan with parchment paper or a silicone baking mat.
In a small saucepan over medium heat, melt the butter. Continue cooking, swirling the pan occasionally, until the butter turns golden brown and develops a nutty aroma, about 3-4 minutes. Remove from heat and stir in the lemon zest, lemon juice, minced garlic, thyme, and rosemary. Set aside.
Place the halved baby potatoes on the prepared sheet pan. Drizzle with 1 tablespoon olive oil and season with 1/4 teaspoon salt and a pinch of black pepper. Toss to coat evenly and spread them out in a single layer.
Roast the potatoes in the preheated oven for 10 minutes.
Remove the sheet pan from the oven and add the green beans. Drizzle the remaining 1 tablespoon olive oil over the green beans and season with a pinch of salt and pepper. Toss gently to combine with the potatoes.
Place the salmon fillets skin-side down on the sheet pan among the vegetables. Brush each fillet generously with the lemon brown butter sauce.
Return the sheet pan to the oven and roast for an additional 10 minutes, or until the salmon is cooked through and flakes easily with a fork and the vegetables are tender.
Remove from the oven. Spoon any remaining lemon brown butter from the pan over the salmon and vegetables before serving.