Trim about 1⁄2" from the tops and bottoms of oranges; quarter oranges lengthwise. Bring 6 cups water to a boil in a 4-qt. saucepan; add oranges. Bring water back to a boil; drain. Repeat boiling process twice more with fresh water. Put oranges, 1 cup sugar, and 4 cups water into a 4-qt. saucepan over medium-high heat. Cook, stirring often, until sugar dissolves and orange rind can be easily pierced with a fork, about 30 minutes. Remove pan from heat and let cool to room temperature.
Heat oven to 350°. Grease a 9" round cake pan with butter and dust with flour; line pan bottom with parchment paper cut to fit. Set pan aside. Whisk together flour, baking powder, baking soda, and kosher salt in a medium bowl and set aside. Remove orange quarters from syrup, remove and discard any seeds, and put oranges into the bowl of a food processor. Pulse until oranges form a chunky purée, 10 to 12 pulses. Add remaining sugar, reserved flour mixture, vanilla, and eggs and process until incorporated, about 2 minutes. Add olive oil; process until combined. Pour batter into prepared pan; bake until a toothpick inserted in center comes out clean, 35–40 minutes. Let cool for 30 minutes.
In a small bowl, whisk orange juice and confectioners' sugar to make a thin glaze. Remove cake from pan and transfer to a cake stand or plate. Using a pastry brush, brush orange glaze over top and sides of cake; let cool completely. Garnish cake with salt.