Heat about 1/2 inch of oil in a large, heavy skillet over medium-high heat; fry plantains until beginning to brown on both sides, 5 to 7 minutes. Transfer to a paper towel-lined plate to cool completely, at least 10 minutes.
Preheat oven to 300 degrees F (150 degrees C). Heavily grease a 9x13-inch cake pan.
Beat egg whites in a bowl using an electric mixer until stiff peaks form. Slowly add 1 cup plus 1 tablespoon white sugar; beat on high until completely smooth and thickened. Add vanilla extract and mix well. Add egg yolks, 1 at a time, and beat until completely smooth again.
Sift flour, baking powder, and salt together in a bowl. Gradually mix flour mixture into egg mixture until batter is just mixed.
Spread 1/2 of the batter into the bottom of prepared cake pan. Place cooled plantains in an even layer on top of batter; top with enough queso fresco to lightly coat. Add 1 more layer plantain and cover with remaining batter, carefully working out any air bubbles with a spatula.
Place pan on the middle rack of preheated oven and bake until the center starts to firm up, 40 to 50 minutes.
You can play with the amount of sugar, omit the cheese, or even use Swiss cheese instead, which sounds strange but pairs quite well with the plantain.
To my friends in Germany: you can use Feneta cheese when queso fresco is unavailable.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.