Add cubed purple sweet potatoes to a pot of salted water. Boil for 15–20 minutes until tender. Drain and let cool slightly.
In a large bowl, mash the potatoes until smooth. Stir in milk, maple syrup, butter, vanilla, cinnamon, nutmeg, and salt. Blend until creamy and cohesive.
Preheat oven to 350°F (175°C). Transfer the mash into a greased baking dish. Smooth the surface with a spatula and bake uncovered for 25 minutes.
In a small saucepan, combine sugar, maple syrup (or honey), and water. Heat over medium until the mixture reaches 240°F (soft-ball stage) on a candy thermometer.
While it heats, beat egg whites, cream of tartar, and salt in a stand mixer on medium-high until soft peaks form.
With the mixer running, slowly stream in the hot syrup. Continue whipping for 5–7 minutes, until thick, glossy peaks form. Add cinnamon and vanilla; beat 30 seconds more.
Spoon the fluff over the baked casserole, forming peaks with the back of a spoon. Use a kitchen torch to toast until golden brown on the tips.
Let it sit for about 10 minutes before serving so the layers can settle slightly.