Preheat oven to 350 degrees F.
For cookie layers: In a bowl knead 1/2 of the cookie dough with 1/4 cup flour until flour is incorporated. Roll dough to 1/4-inch thickness on a lightly floured surface. Using the springform pan as a template, cut 1 large cookie. Repeat with remaining dough and 1/4 cup of flour. Place both large cookies on baking sheets or plates and cover with plastic wrap. Place in the freezer and chill for 15 minutes. Remove from freezer, place on cookie sheet and bake for 15 to 18 minutes, until golden brown.
For pudding: In a bowl with an electric hand mixer, beat pudding and milk until thickened, about 2 minutes.
To assemble: In the bottom of a 9-inch springform pan, place 1 of the cookies. Over the top of the cookie, spread the softened ice cream. Top with sliced bananas. Layer with another cookie. Top with banana pudding and smooth. Sprinkle with toasted almond slices. Cover with a layer of plastic wrap and a layer of aluminum foil. Freeze for a minimum of 8 hours, preferably overnight.
For the meringue: With an electric mixer on medium speed, whisk pasturized egg whites to soft peaks. Beat in sugar and almond extract and beat to stiff but not dry peaks. Be careful not to overbeat.
Preheat broiler. Remove dessert from freezer. Remove springform pan by first dampening a kitchen towel with hot water. Wrap towel around springform pan to loosen the dessert. Remove from springform pan and place on an oven-proof platter. Cover with meringue. Place under broiler for no more than 3 minutes until golden peaks form. WATCH CAREFULLY, as it can burn quickly. Serve immediately.
Special equipment: 9-inch springform pan .