1. Place oven rack in middle of oven, and preheat oven to 350°. Melt butter over low heat in a 10-inch (2-inch-deep) cast-iron skillet. Stir in 3/4 cup sugar, combining well. Arrange apple quarters in a circular pattern around the outside of the skillet, and fill in the center with additional quarters. Increase heat to medium, and cook, without stirring, 15 to 20 minutes or until sugar turns golden.
2. Beat egg yolks and 1/2 cup sugar in a large bowl of an electric mixer at high speed about 3 minutes or until thick and pale. Reduce speed to medium, and slowly add olive oil and vanilla until well combined. Remove from mixer, and stir in flour just to combine.
3. Place egg whites and 1/2 teaspoon sea salt in a separate large bowl of an electric mixer, and beat at medium speed until frothy. Add remaining 1/4 cup sugar, and continue beating just until soft peaks form.
4. Fold one-third of the egg whites gently into the yolk mixture. Fold in remaining whites, being careful not to overmix. Pour batter into skillet with apples. Bake at 350° for 25 to 30 minutes or until golden brown and a wooden pick inserted in center comes out clean.
5. Cool cake about 5 minutes, then run a knife around edges to loosen. Invert onto a large platter, and, if desired, sprinkle 1/4 teaspoon sea salt over the apples and caramel. Serve warm with lightly sweetened whipped cream or vanilla ice cream.