Preheat the oven to 350°F (175°C). Grease and flour a 9×13 inch baking pan.
In a medium bowl, sift together the all-purpose flour, unsweetened cocoa powder, baking powder, and salt. Set aside.
In a large bowl, beat the egg yolks with 3/4 cup granulated sugar until the mixture is pale and thick, about 3-4 minutes.
Add the whole milk, vanilla extract, and vegetable oil to the egg yolk mixture and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing gently until just combined.
In a separate clean bowl, beat the egg whites until soft peaks form. Gradually add the remaining 1/4 cup granulated sugar and continue beating until stiff peaks form.
Fold the beaten egg whites gently into the cake batter until fully incorporated, being careful not to deflate the mixture.
Pour the batter evenly into the prepared baking pan and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Using a fork or skewer, poke holes all over the surface of the cake to allow the milk mixture to soak in.
In a medium bowl, whisk together the evaporated milk, sweetened condensed milk, and whole milk until fully combined.
Slowly pour the three-milk mixture evenly over the warm cake, allowing it to absorb the liquid. Refrigerate the cake for at least 4 hours or overnight for best results.
Before serving, prepare the whipped cream topping by beating the heavy whipping cream, powdered sugar, and vanilla extract together until stiff peaks form.
Spread the whipped cream evenly over the soaked cake.
Sprinkle the semisweet chocolate shavings over the whipped cream as garnish.
Cut into squares and serve chilled.