Preheat the oven to 350 degrees F (175 degrees C). Spray a 9x13x2-inch baking pan with cooking spray. Line the bottom with parchment paper.
Whisk cake flour, all-purpose flour, baking powder, salt, and baking soda together in a medium bowl and set aside.
Combine white sugar and lemon zest in a large mixing bowl. Use your fingers to rub the zest into the sugar until fully incorporated. Whisk in olive oil, lemon extract, and vanilla extract. Add eggs and egg yolks, 1 at a time, whisking vigorously between each addition. Add lemon juice and whisk again; add creme fraiche, whisking a final time to incorporate.
Add the flour mixture to the sugar mixture all at once and use a rubber spatula to very gently fold the dry into the wet. Transfer the batter to the prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out mostly clean with a crumb or two, rotating at the halfway point, 30 to 33 minutes. Set on a cooling rack and let the cake cool for 10 to 15 minutes. Invert the cake, right-side up, on to the rack, and set over a rimmed baking sheet.
While cake cools, make the glaze. Place confectioners’ sugar and lemon juice in a large bowl. Whisk vigorously until smooth. Gently pour over the cake, using an offset spatula to spread the glaze if it pools, and let the cake cool to room temperature. You can poke the cake with a toothpick to help the glaze seep in.