Beat cream in a chilled glass or metal bowl using an electric mixer until frothy. Add sugar gradually, continuing to beat until soft peaks form. Mix in strawberry extract. Spread a little strawberry cream on the bottom of a rectangular dish.
Pour milk into a bowl and dip each side of the Oreo into the milk for a couple of seconds then shake off excess.
Sandwich cookies together with a heaped teaspoon of strawberry cream each. Lay them into the dish, working lengthwise about 10 across the dish. Continue laying cookies into the dish to create about 4 rows.
Cover the top and sides of the cookies with the rest of the cream. Sprinkle grated chocolate on top.
Place a sheet of parchment paper over the top of the cake to prevent it from sticking. Cover icebox cake and dish with aluminum foil.
Refrigerate until cookies have turned soft, at least 8 hours to overnight. Garnish with strawberries before serving.