Preheat oven to 350F and cut a circle of parchment paper for the bottom of a 10" spring form pan. Prepare the vermouth soak by combining the coffee and vermouth, set aside.
Cake; beat the eggs for 2 minutes on medium speed. Add the sugar and beat on high for 10 minutes, or until the batter has tripled in volume. (The longer the batter is beaten, the better the result). Blend in the oil just until blended. Gently fold in the flour and baking powder just until the batter is free of any lumps, do not over mix.
Pour the batter into the prepared pan and bake for 30 minutes, or until the cake tester comes out clean. Cake will fall slightly once removed from the oven. Leave the cake in the pan until completely cool.
Cut the cooled cake into 2 layers and brush all of the vermouth soak on both of the layers on the cut sides. Using the top of the cake as the bottom layer (the flat bottom for the top), spread the bottom layer with the lemon custard, then top with the second layer. Frost the entire cake with whipped cream.
LEMON CUSTARD; Prepare the whipped topping mix with 1/2 cup of milk and vanilla extract according to the package directions. Prepare the pudding according to the package but using only 1 cup milk. Fold the pudding into the topping mix until completely incorporated.
SWEETENED WHIPPED CREAM; Beat the whipping cream with the 6 tablespoons of sugar until firm peaks are formed when the beaters are lifted, do not overbeat. Refrigerate until ready to use.