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.