1 cup Arbequina extra-virgin olive oil, plus more for pan
1/2 cup plus 2 tsp. granulated sugar, plus more for pan
2 sweet apples
1 tsp. ground cinnamon
1/2 tsp. freshly ground black pepper
1/2 tsp. ground cardamom
1/2 tsp. ground ginger
1 1/2 cups all-purpose flour
2 tsp. baking powder
2 large eggs
2 egg yolks
1/2 cup brown sugar
1 tsp. vanilla extract
1/2 tsp. kosher salt
Preheat oven to 350°F. Pour about 1 Tbsp. Arbequina EVOO into a 9x5" loaf pan and coat bottom and sides with fingertips. Sprinkle granulated sugar onto sides and bottom of pan to lightly coat. Grate apples, peel included, with the large holes of a box grater. Gather apples in a clean kitchen towel and wring out excess liquid. You want them as dry as possible when they go into the batter. Whisk cinnamon, pepper, cardamom, and ginger in a medium bowl. Transfer 1/4 tsp. spice mixture to a small bowl; set aside. Add flour and baking powder to remaining spice mixture and whisk to combine. Whisk together eggs, egg yolks, brown sugar, vanilla, salt, and 1/2 cup granulated sugar in a large bowl. Continue whisking vigorously until lightened in color and thickened. Whisking with one hand and pouring with the other, slowly stream 1 cup Arbequina EVOO into egg mixture until combined. Gently fold flour mixture into whipped eggs with a flexible rubber spatula just until combined, taking care not to over mix. Fluff up grated apples and scatter over batter. Gently fold to incorporate.
Transfer batter to prepared pan; smooth top. Add 2 tsp. granulated sugar to reserved spice mixture and mix to combine. Sprinkle spiced sugar evenly over top of batter.
Bake cake until deep golden brown and a tester inserted into the center comes out clean, 60–70 minutes. Let cake cool in pan. Run a butter knife or offset spatula along edge of cake to loosen and invert onto a platter.