Preheat the oven to 350°F (175°C). Grease and flour a 9×13-inch baking dish.
In a medium bowl, whisk 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 of granulated sugar until the mixture is pale and thick, about 3 minutes.
Add the whole milk, vegetable oil, and 1 teaspoon vanilla extract to the egg yolk mixture and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
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.
Gently fold the egg whites into the cake batter in three additions, being careful not to deflate the mixture.
Pour the batter into the prepared baking dish and smooth the top.
Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
While the cake is still warm, poke holes all over the top using a fork or skewer.
In a large bowl, whisk together the evaporated milk, sweetened condensed milk, and whole milk.
Slowly pour the milk mixture evenly over the cake, allowing it to soak in.
Refrigerate the cake for at least 4 hours or overnight to absorb the milk mixture fully.
Before serving, prepare the whipped cream by beating the heavy cream, powdered sugar, and 1 teaspoon vanilla extract until stiff peaks form.
Spread the whipped cream evenly over the soaked cake.
Sprinkle the semisweet chocolate shavings on top as a garnish.
Cut into squares and serve chilled.