1. Start by peeling and slicing the sweet potatoes into rounds, about 1/2 inch thick. This uniform thickness ensures even cooking.

2. Melt the unsalted butter in a large skillet over medium heat, then add the sweet potato slices, arranging them in a single layer.

3. Pour the brown sugar and water over the sweet potatoes, then sprinkle with the cinnamon, nutmeg, and salt. Gently stir to coat the sweet potato slices evenly.

4. Reduce the heat to low, cover the skillet with a lid, and let the sweet potatoes cook until they are fork-tender and beautifully glazed, about 35-45 minutes. Be sure to stir occasionally to prevent sticking and ensure even cooking.

5. Once the sweet potatoes are tender, remove the lid and continue cooking for an additional 5 minutes to allow the glaze to thicken and the sweet potatoes to caramelize to perfection.

6. Remove the skillet from heat and stir in the vanilla extract. The glaze will continue to thicken as it cools.

7. Spoon the irresistible brown sugar-coated sweet potatoes onto a serving dish and enjoy their warm, comforting, and utterly delightful flavors.