Heat the oven to 350 degrees F. Put the cranberries and 1 cup of the orange juice in a small bowl and let sit at room temperature for at least 30 minutes.

Spread the walnuts in an even layer on a rimmed baking sheet and toast in the oven until golden brown, about 12 minutes. Transfer the nuts to a plate to cool. When cool, crumble the nuts into smaller pieces.

Increase the oven temperature to 450 degrees F. Toss the diced sweet potato with the maple syrup, the remaining 1/4 cup orange juice, the 1/2 teaspoon salt, and the black pepper. Spread the potato pieces on the same rimmed baking sheet you used to toast the nuts and roast, stirring occasionally, until just tender, 10 to 15 minutes. Set aside

Meanwhile, bring 4 quart water and the 1/4 cup salt to a boil in an 8-quart pot. Put the beans in the boiling water, stir once or twice to distribute them, and cook until just tender throughout, 4 to 5 minutes, depending on their size and freshness. Drain the beans well in a colander.

When ready to serve, strain the cranberries and reserve the juice. Melt the butter in a large (preferably 12-inch) skillet over medium-high heat until bubbling but not browned. Add the chopped sage and let sizzle for just a few seconds, stirring, to flavor the butter and crisp the sage. Add the green beans and spread them out in the pan. Let them cook for 2 to 3 minutes without stirring. Add the cooked sweet potatoes, toasted walnuts, and strained cranberries. Toss all together and cook, stirring occasionally, for 2 minutes more, to let the ingredients warm and their flavors mingle. Add the reserved orange juice, stir well, and cook until most of the liquid has evaporated, about 1 minute more. Taste and season with salt and pepper as needed. Transfer to a serving dish and serve immediately.

You can make this recipe up to 4 hours ahead, stopping after you par-cook the green beans. Plunge the hot green beans into an ice bath to stop the cooking, drain well in a colander, and set aside until ready to proceed with the recipe.