Heat the oven to 400°F with a rack positioned in the middle. Mist an 8-inch square baking pan with cooking spray; line the bottom with a square of kitchen parchment and mist the parchment.

In a 12-inch nonstick skillet over medium-high, cook the 28 grams (2 tablespoons) butter, stirring, until the butter begins to brown, about 30 seconds.

Add the apples, stir to coat, then distribute in an even layer. Cook without stirring until beginning to brown, about 2 minutes.

Using a silicone spatula, stir and flip the apples, then redistribute in an even layer. Cook, stirring occasionally, until the apples are tender and translucent, about another 2 minutes.

Off heat, add the brown sugar and stir gently until melted and combined with the apples. Transfer the mixture to the prepared pan and gently press into an even layer that covers the entire bottom; set aside.

In a stand mixer with the whisk attachment, beat the egg whites on medium-high until frothy, about 20 seconds.

With the mixer running, add the salt, then gradually add 54 grams (¼ cup) of white sugar. Continue to beat until the whites hold soft peaks, about 2 minutes.

Transfer the whites to a medium bowl; reserve the mixer bowl and attachment (no need to wash them).

In the now-empty mixer bowl, combine the remaining 170 grams (12 tablespoons) butter and the remaining 160 grams (¾ cup) white sugar. Using the whisk attachment, beat on medium-high until lightened and fluffy, about 3 minutes, scraping the bowl once or twice.

With the mixer running on medium-low, add the yolks one at a time, beating until incorporated, about 20 seconds, after each addition. Stop the mixer and scrape the bowl. Add the ricotta and vanilla, then beat on medium-low just until incorporated, 25 to 30 seconds; do not overmix (the mixture will not be smooth).

Remove the bowl from the mixer and fold by hand with the spatula, scraping the bottom and sides of the bowl, to ensure the mixture is well combined.

Sprinkle the semolina over the ricotta mixture and fold with the spatula until incorporated. Scoop about ½ cup of the whipped egg whites onto the mixture and fold them in to lighten and loosen the base. Scrape on the remaining whites, then gently fold until no white streaks remain.

Scrape the batter on top of the apples and, using a small offset spatula, gently spread into a smooth, even layer.

Bake until golden brown and the center is just slightly wobbly when the pan is gently jiggled, 30 to 35 minutes.

Cool in the pan on a wire rack to room temperature, 1½ to 2 hours. Run a paring knife around the inside edge of the pan to loosen the cake, then invert onto a platter. Lift off the pan and peel off the parchment.