Pour the cream into the bowl of an electric stand mixer. Add the sugar, 6 tablespoons Kahlua, and vanilla extract. Using the whip attachment, whip until firm peaks form. Add 1/2 cup shredded coconut and mix to combine.
Pour the remaining 8 tablespoons of Kahlua into a shallow dish. Make a large 22-inch square piece of plastic wrap by pressing two sheets together in the middle. Then line a 9x5-inch bread loaf with the sheet of plastic wrap, so the edges hang over on all sides. Spread a thin 1/4- to 1/3-inch layer of whipped cream over the bottom of the pan.
Place 4 graham crackers in the dish of Kahlua. Flip so both sides are coated. Then trim the edge of each cracker so they can fit into the bottom of the pan side-by-side. Press until the crackers are even. Spread another 1/4 - 1/3 inch layer of whipped cream over the crackers.
Repeat dipping and layering the chocolate graham crackers, with even layers of cream in between, until you have 5 layers of graham crackers in the pan. Pull the plastic wrap up on the sides as needed. Spread the top of the pan with a level layer of cream. Wrap the plastic over the top and freeze for at least two hours.
Cover the remaining whipped cream and place in the refrigerator. Meanwhile, measure 1/2 cup shredded coconut into a skillet. Set over medium heat and lighting toss until the coconut is toasted. Watch carefully so it doesn't burn. Turn off the heat and cool.
Use a vegetable peeler to shave the dark chocolate for a garnish.
Once the icebox cake is frozen solid, unwrap the top and flip the cake onto a platter. Remove the plastic. Then spread the remaining whipped cream over the side and top to cover the creases. Sprinkle the top with toasted coconut and chocolate shavings. Then place back in the freezer until ready to serve. If making more than 12 hours ahead, cover with plastic once set, to avoid freezer burn.
NOTE: If the crackers break as you are layering them into the loaf pan, don't worry! Just puzzle-piece them into the pan. Once they absorb the Kahlua and plump up, you will never see the cracks.