Preheat oven to 350 degrees. Line a 9 x 13 inch cake pan with parchment paper.
To make the hot fudge sauce, whisk together the dark chocolate chips, coconut milk, cocoa powder, Kahlua (if using), vanilla, and a pinch of salt until melted and smooth. Remove from heat and let cool.
In a bowl, beat together the coconut oil, yogurt, and sugar until fluffy, about 2 minutes. Beat in the eggs, one at a time, then the vanilla. Add the flour, baking powder, baking soda, and salt. Whip until just combined. Stream in the coconut milk until incorporated. Fold in the crushed Oreos.
Pour the batter into the prepared cake pan. Dollop 1/2 cup of the hot fudge sauce over the batter and swirl with a butter knife in a figure-8 motion. Bake 30-35 minutes until the center is set and an inserted toothpick emerges clean. Cool in the pan 20 minutes, then carefully lift the parchment and set the cake on a wire rack. Set in the fridge to cool completely, about 30 minutes.
Heat the white chocolate chips in the microwave for 1 minute. Stir until completely melted. Cool 5 minutes. For the frosting, whip together the powdered sugar, coconut oil, white chocolate, vanilla, and 2-3 tablespoons of coconut milk until fluffy. Fold in the Oreo pieces. Set the frosting in the fridge for 5 minutes.
Remove the cooled cake from the fridge. Spread a thin layer of the fudge sauce overtop, then spread the Oreo frosting overtop. Swirl a few additional spoonfuls of fudge sauce into the frosting. Set the cake in the fridge until ready to serve.