Preheat oven to 350°F.
Line a sheet tray with paper towels. Using a mandoline or really sharp knife, cut the oranges into round slices about 1/4-inch thick. Place orange slices onto the paper towel-lined sheet tray. Once the sheet tray is full, cover with another layer of paper towels and place the rest of the oranges on top. Place another layer of paper towels over the last few slices of orange and gently press to dry them out a bit. Set aside.
In a medium mixing bowl, whisk together the eggs, olive oil and 1 cup of sugar. Mix until completely smooth. In another medium bowl, mix together flour, baking powder and salt. Add the dry ingredients into the wet and, using a spatula, fold until the batter has no more dry spots of flour. Set aside.
Place a 10-inch cast-iron skillet over high heat and add the butter. Pour the remaining 1/2 cup sugar onto a plate. Remove the paper towels from the oranges and, working quickly, coat both sides of the orange slices with sugar, shaking off any excess, then place into the hot skillet. Completely cover the bottom of the pan and place any extra orange slices over them. Once all the orange slices are in the pan, cook for minute or two or until the sugar begins to get bubbly on the sides. Carefully add in the shot of orange liqueur—it will most likely ignite, so be prepared. Cook for a few minutes or until the flame dies down. Pour the cake batter into the center of the pan directly over the oranges. Spread the cake batter out slightly—it will spread out in the oven so you don’t need to spread it too much.
Place the pan into the oven and bake for 35 minutes.
Cool for 15 minutes before inverting onto a plate then set aside for a few hours before slicing into it.MORE
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