Preheat the oven to 350 degrees F (175 degrees C). Generously grease a 10x15-inch baking dish with shortening.

Beat together cake mix, cold water, eggs, oil, and 1 teaspoon vanilla in a large bowl with an electric mixer until very smooth, scraping down the sides of the bowl as needed. Pour batter into the prepared baking dish. Smooth the top, shake the dish, and tap lightly on the counter 2 or 3 times to eliminate air bubbles.

Bake in the preheated oven until a toothpick inserted into the center of cake comes out clean, about 22 minutes. Let cake cool in the dish for at least 30 minutes. Poke holes all over cake with a bamboo skewer or the handle of a wooden spoon.

Whisk together condensed milk, evaporated milk, and half-and-half in a large bowl until well combined; slowly pour milk mixture all over the top of cake and allow to soak in. Wrap cake with plastic wrap and refrigerate for at least 3 hours to overnight.

Beat together cream, sugar, and remaining 1/2 teaspoon vanilla in a medium bowl with an electric mixer until fluffy; spoon over cake slices to serve.