1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.