Cut the cookies to form 1-by-1-by-1/2-inch squares. Line a 9-inch springform pan with plastic wrap. Arrange the cookie squares all the way around the sides of the pan with the color lines running perpendicular to the bottom of the pan. Repeat with an additional ring of cookies above the first ring, lining up the colors as much as possible, so the cookies come 2 inches up the sides of the pan and almost completely cover them.
Lay one cookie square in the middle of the bottom of the pan. Arrange a second cookie square next to it, with the color lines perpendicular to the color lines in the first cookie. Continue arranging cookie squares to create a parquet floor pattern. Fill the bottom of the pan with cookies, cutting some to fill in any odd-shaped areas. Chill until cold, about 30 minutes.
Combine the chocolate and oil in a microwave-safe bowl and microwave in 30-second intervals, stirring between each, until melted, about 2 minutes. Using a pastry brush, coat the cookies on the bottom and sides with a thin layer of chocolate. Chill until set, about 5 minutes.
Spread one of the pints of vanilla ice cream in an even layer over the chocolate-coated bottom cookies. Freeze until hard, about 1 hour. Spread the chocolate ice cream in an even layer over the vanilla. Freeze until hard, about 1 hour. Spread the remaining pint of vanilla ice cream in an even layer over the chocolate ice cream. Cover with the remaining cookie pieces. Freeze until solid, 6 hours to overnight.
When ready to serve, invert the cake onto a plate so the bottom layer of cookies is on the top. Remove the pan and peel off the plastic wrap. Cut with a hot dry knife.
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