Place the peeled and cubed sweet potatoes in a medium saucepan and add enough cold water to cover them by an inch. Bring to a boil over high heat.
Reduce the heat to medium-low, cover, and let simmer for 20 to 30 minutes, or until the potatoes are very tender when pierced with a fork. Drain the potatoes thoroughly.
Preheat your oven to 350°F (175°C).
Transfer the hot, drained sweet potatoes to a large bowl. Mash them with a potato masher or an electric mixer on low speed until smooth.
Add the melted butter, evaporated milk, brown sugar, beaten eggs, cinnamon, vanilla, and nutmeg to the mashed sweet potatoes. Mix until the filling is smooth and well combined, but do not overmix.
Pour the filling into the unbaked pie crust and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until the edges are set and the center jiggles only slightly. A thermometer inserted in the center should read 160°F (71°C).
Let the pie cool completely on a wire rack for at least 2 hours. This allows the filling to set properly before slicing and serving.