2 medium sweet potatoes, pierced with the tines of a fork
1/4 teaspoon ground ginger
1/2 teaspoon pure vanilla extract
1/4 teaspoon salt
4 large eggs, separated
2 tablespoons packed light-brown sugar
1 cup whole milk
1 2-inch piece peeled fresh ginger, thinly sliced
1/4 cup unsalted butter, plus 1/2 cup melted, plus more for pan
1/4 cup all-purpose flour
1/3 cup, plus 1/4 cup granulated sugar, plus more if needed
1/4 teaspoon ground cinnamon, plus more if needed
9 sheets phyllo dough (17 by 12 inches), thawed if frozen
Pinch of cream of tartar
Preheat oven to 400 degrees. Bake potatoes until tender, 1 hour to 1 hour 15 minutes. Remove from oven; let stand until cool enough to handle.
Peel potatoes, and press through a ricer into a large bowl (you should have about 1 cup); let cool completely. Stir in ground ginger, vanilla, salt, egg yolks, and brown sugar; set aside.
Meanwhile, heat milk and fresh ginger in a medium saucepan over medium heat until just under a boil. Remove from heat; let stand 30 minutes. Pour through a fine-mesh sieve into a bowl.
Melt 1/4 cup butter in a medium saucepan over medium-high heat. Whisk in flour; cook, whisking, 1 minute. Whisk in milk mixture. Bring to a boil. Reduce heat to medium, and cook 1 minute. Stir into potato mixture.
Butter a 9-inch springform pan, and place on a baking sheet; set aside. Stir 1/3 cup granulated sugar and the cinnamon in a small bowl. Brush 1 phyllo sheet with melted butter. Sprinkle lightly with cinnamon-sugar mixture. Fold in half crosswise; brush with butter. Sprinkle lightly with cinnamon-sugar mixture, and fit into prepared pan, folded side in, allowing a 2 1/2-inch overhang. Repeat, overlapping sheets to cover bottom.
Put egg whites into the bowl of an electric mixer fitted with the whisk attachment; beat on high speed until foamy. Add cream of tartar; beat until soft peaks form. With machine running, gradually add remaining 1/4 cup granulated sugar; beat until stiff glossy peaks form. Whisk one-third of the egg whites into potato mixture. Gently but thoroughly fold in remaining egg whites.
Pour over phyllo; fold overhang over filling. Sprinkle with cinnamon-sugar mixture (if needed, combine 1 to 2 tablespoons more sugar and a pinch of cinnamon). Reduce oven temperature to 375 degrees. Bake pie until puffed and just set in center, 45 to 50 minutes. Remove from oven; let stand until slightly cooled and center has fallen, about 20 minutes.
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