Heat oven to 350°F.
Line 1-quart round bowl with plastic wrap so there is enough overhang to cover top. Scoop one flavor of ice cream into bowl and pack, smoothing top evenly. Top with second flavor of ice cream, smoothing top. Bring sides of plastic wrap up and cover top of ice cream. Freeze several hours or overnight or until very firm.
Tear off 10-inch wide sheet of aluminum foil. Place on large baking sheet; grease 8-inch circle in center.
Work together cookie dough and walnuts with wooden spoon in large bowl until well blended. (If dough becomes too soft to work with, refrigerate to firm, about 10 minutes.) Shape dough into ball; place in center of greased foil. Pat with hands or roll with floured rolling pin into 8-inch circle.
Bake in 350°F oven 12 to 15 minutes or until lightly browned at edges. Remove cookie, still on foil, to wire rack to cool completely. Increase heat to 475°F.
Remove bowl with ice cream from freezer; let stand until ice cream is softened enough to lift with plastic wrap from bowl; it may be necessary to loosen sides by gently running long thin metal spatula between bowl and plastic wrap around edge. (Also, to speed loosening, you may dip bottom half of bowl into large bowl of warm water for a few seconds.)
Stir dried egg whites into warm water in small bowl to dissolve completely. Beat at high speed with electric mixer until fluffy. Gradually add sugar, beating until meringue forms stiff peaks.
Loosen cookie from foil, then return to same piece of foil. Place on cool, insulated baking sheet. Lift ice cream from bowl. Fold sides of plastic wrap back from top of ice cream; invert ice cream onto center of cookie. Peel off plastic wrap. Completely cover ice cream with meringue, swirling to within 1/4 inch of edge of cookie crust.
Bake in 475°F oven 2 to 4 minutes or until meringue is lightly browned. Using wide spatula, gently slide Alaska from foil onto serving platter. Serve immediately.