Preheat oven to 350°F/175°C. Line a 9x13" baking dish with parchment paper.
In a large mixing bowl, whisk together milk, vinegar, sugar, and vanilla. Let sit for about 5 minutes to dissolve the sugar, then whisk in oil.
In a separate bowl, sift flour, cornstarch, salt, and baking soda and whisk together.
Sift dry ingredients over wet and whisk together just until batter is smooth. Be careful not to over-mix or your cake can turn out dense.
Transfer batter to prepared baking dish. Tap the baking dish on the counter a few times to remove air bubbles. Bake for 25-30 minutes, until a toothpick inserted in the center comes out clean. You want to just slightly over-bake the cake so it dries out a little, since we will be dipping it in coffee later.
Place the dish on a wire rack. Let the cake cool in the pan for 10 minutes, then remove the cake from the baking dish. Carefully peel off the parchment paper and let the cake cool on the wire rack.
Cream
Prepare the cream while the cake is baking (or make it the day before).
Add tofu, sugar, vanilla extract, and lemon juice to a blender or food processor and blend until smooth.
Scoop out the thick top layer of the coconut cream, leaving behind any liquid. Add to food processor and pulse a few times to combine - be careful not to over-blend.
Transfer the cream to a bowl and tap on the counter a few times to remove air bubbles. Cover and store in the fridge for a few hours to thicken.
Assemble
Cut cake in half lengthwise and cut into strips (or "fingers"). You can also cut the cake in half widthwise if it baked up tall (like if you used a smaller pan).
You can use an 8x8" or 8x10" dish, or whatever will fit two layers of the cake fingers and cream.
Briefly dip each cake finger into the coffee on both sides, then place on the bottom of the baking dish in a single layer.
Spread half the cream on top in a thick layer.
Repeat with remaining cake fingers and cream.
Cover and refrigerate overnight. Use a sieve or tea strainer to dust the top with cocoa powder just before serving.
Notes
Coconut Cream: Chill a can of coconut cream or full-fat coconut milk in the fridge at least the day before. Measure out only the thick cream layer, not including any liquid.To store: Cover the cake with plastic wrap and store in the fridge for up to 4 days. I haven't tried freezing it, but I'm afraid the cream layer might separate or get an icy texture when thawed.