1 1/2 tablespoons unsalted butter, cut into small pieces
Preheat an oven to 325 degrees F. Position a rack in the center of the oven. Position a second rack directly below it.
FOR THE CRUST: In a large bowl, combine the graham cracker and gingersnap crumbs, sugar, and ginger. Stir in the melted butter. Firmly press the crumbs into the bottom of a 9 1/2-inch springform pan. Brush the crust lightly with the egg white and chill for at least 15 minutes before filling and baking.
FOR THE FILLING: In a large bowl, beat together the cream cheese, sour cream, and brown and granulated sugars, using an electric mixer, until smooth. Gently beat in the egg yolks, then the pumpkin puree, vanilla, cornstarch, ginger, cinnamon, nutmeg, and cloves. In a large bowl, beat the egg whites, with a whisk or an electric mixer, until stiff peaks are formed. Fold the egg whites into the cream cheese mixture.
Pour the filling into the prepared pan and place in the oven. Set an 8-inch-square baking pan filled halfway with hot water on the rack below the cheesecake (this helps prevent the top of the cheesecake from cracking). Bake until the center does not tremble when the cake is gently shaken, about 45 minutes to an hour.
Remove from the oven and allow to cool for several minutes. Place a rack 4 to 5 inches from the heat source of a broiler. Preheat the broiler.
FOR THE TOPPING: in a small bowl, combine the walnuts, ginger, and sugar. Add the butter and mix together with a fork or clean fingers until crumbly. Sprinkle over the warm cheesecake and place in the broiler to melt the sugar and caramelize the topping, 30 to 40 seconds. Watch carefully so that the topping does not burn.
Remove the cheesecake from the broiler and let cool to room temperature. Cover and chill for at least 6 hours or as long as overnight. When it is chilled, run a knife around the edge of the cheesecake and remove the pan rim. Set the cheesecake on a serving plate without removing the pan bottom.