Prepare the sweet potatoes by peeling them and slicing them into uniform 1/2-inch thick rounds to ensure even cooking.
In a large, heavy-bottomed skillet or Dutch oven (wide enough to hold potatoes in mostly one layer), melt the butter over medium heat.
Add the granulated sugar, brown sugar, water, cinnamon, nutmeg, and salt to the melted butter. Stir constantly until the sugars have dissolved and the mixture begins to bubble.
Add the sweet potato rounds to the skillet, tossing gently to coat them thoroughly in the syrup.
Reduce the heat to low and cover the skillet with a tight-fitting lid. Simmer for 35 to 45 minutes, or until the potatoes are fork-tender but not falling apart. Stir gently every 15 minutes to baste the potatoes.
Remove the lid and stir in the vanilla extract. Increase the heat slightly to medium-low and simmer uncovered for another 5-10 minutes. This allows the liquid to reduce into a thick, sticky glaze.
Remove from heat and let the dish sit for 5 minutes before serving to allow the sauce to thicken further. Spoon the extra glaze over the potatoes when plating.