Cut the sweet potatoes crosswise into 1-inch-thick rounds. Halve or quarter any extra-large pieces so that they are all similar in size.
Melt the butter in a large Dutch oven over medium heat. Combine the brown sugar, granulated sugar, pumpkin pie spice and salt in a small bowl. Add the sugar mixture to the Dutch oven and whisk until it forms a paste. Add the potatoes and stir well, spooning the sugar mixture over the potatoes to coat evenly.
Cover and cook the potatoes, stirring occasionally and gently to avoid breaking them up, until they are just tender when pierced with the tip of a knife and the sauce resembles a thin syrup, 25 to 30 minutes.
Meanwhile, preheat the oven to 350 degrees F. Spread the pecans on a baking sheet and toast until dark brown with a light scent of roasted nuts, about 8 minutes. Season the pecans with some salt while warm, let them cool completely and coarsely chop.
Transfer the potatoes with their sauce to a large serving platter and sprinkle the pecans over the top.
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