Preheat the oven to 400˚F. Oil a 9-by-13-inch baking dish. For the glaze: Combine the apple cider, brown sugar, soy sauce, ginger, paprika, and 2 grated garlic cloves in a medium saucepan over medium-high heat. Bring to a boil, reduce to a simmer , and cook, stirring often, until syrupy, about 15 minutes. Whisk in 2 tablespoons butter and the juice of ½ lemon. Set aside half of the glaze in a small bowl for serving.

Pat the salmon dry with paper towels and place in the prepared baking dish. Season with ¾ teaspoon salt and ¼ teaspoon pepper, then brush with the remaining glaze in the saucepan. Roast until the salmon is opaque and flakes easily with a fork, 18 to 20 minutes.

Meanwhile, bring 1 inch of water to a simmer in a large pot fitted with a steamer basket. Add the potatoes, cover, and cook for 8 minutes. Add the green beans on top of the potatoes, cover, and cook until the beans are crisp-tender and the potatoes are easily pierced with a paring knife, about 5 minutes.

Carefully remove the steamer basket and vegetables from the pot. Wipe the pot clean, then add the remaining 4 tablespoons butter and melt over medium heat. Stir in the remaining 3 grated garlic cloves and cook until fragrant, about 30 seconds. Return the potatoes and green beans to the pot. Toss with the juice of the remaining ½ lemon, 1¼ teaspoons salt, and ½ teaspoon pepper.

Brush the salmon with the reserved glaze and serve with the vegetables.