Preheat the oven to 300 degrees . In a medium saucepan, combine the potatoes and enough salted water to cover by 1 inch. Bring to a boil, then lower the heat and simmer until just tender, about 10 minutes. Increase the heat to medium-high, add the broccoli and cook until tender, about 4 minutes. Drain, then mash the broccoli and potatoes with 2 tablespoons butter. Season with salt and pepper; cover to keep warm.
Season the salmon fillets with salt and pepper, then arrange skin side up, in a single layer, in a greased, foil-lined large baking dish. Roast until cooked through, about 10 minutes for medium. Discard the skin.
In a small skillet, melt the remaining 6 tablespoons butter over medium heat. Add the scallion whites and cook, swirling the pan, until the butter is lightly browned, about 5 minutes. Remove from the heat and stir in the scallion greens and lemon juice; season with salt and pepper. Spoon the sauce over the fish and serve with the mashed broccoli potatoes.