Preheat the oven to 400°F (200°C).
In a large bowl, toss the halved baby potatoes, baby carrots, and trimmed green beans with 2 tablespoons olive oil, 1 teaspoon dried thyme, 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper until evenly coated.
Spread the vegetables evenly on a large sheet pan in a single layer.
Place the sheet pan in the oven and roast the vegetables for 15 minutes.
While the vegetables roast, prepare the lemon brown butter sauce. In a small saucepan over medium heat, melt 4 tablespoons unsalted butter.
Continue cooking the butter, swirling occasionally, until it turns a golden brown color and develops a nutty aroma, about 3 to 4 minutes. Be careful not to burn it.
Remove the saucepan from heat and stir in the minced garlic, lemon zest, and lemon juice. Set aside.
After the vegetables have roasted for 15 minutes, remove the sheet pan from the oven.
Push the vegetables to the sides of the pan to make space in the center for the salmon fillets.
Place the salmon fillets skin-side down in the center of the sheet pan.
Season the salmon with 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper.
Pour the lemon brown butter sauce evenly over the salmon fillets and vegetables.
Return the sheet pan to the oven and bake for an additional 5 to 7 minutes, or until the salmon is cooked through and flakes easily with a fork.
Remove from the oven and let rest for 2 minutes before serving.
Serve the salmon fillets with the roasted vegetables, spooning any remaining lemon brown butter sauce from the pan over the top.