Wash the oranges well under warm running water, scrubbing the skins gently to remove any wax or residue. Place the whole oranges in a large stockpot and add enough water to cover them by about 2 inches.
Bring the water to a boil over high heat, then reduce to a gentle simmer. Cover and cook for 60 minutes, turning the oranges every 15 minutes if they bob to the surface, until the skins are very soft and a knife slides in with almost no resistance.
Lift the oranges out with a slotted spoon and let them cool on a plate for 20 to 30 minutes. Cut each orange into quarters, pick out every seed, and trim off the hard stem nub, keeping all the peel, pulp, and juice.
Position an oven rack in the center of the oven and preheat to 350°F (175°C). Grease a 9-inch springform pan, line the base with a parchment circle, grease the parchment, and close the ring securely.
Add the quartered oranges and any juices to a food processor and process for 1 to 2 minutes, scraping down the sides, until you have a thick, mostly smooth puree. You should have about 1 3/4 cups.
Crack the eggs into a large mixing bowl, add the sugar, and whisk vigorously for 2 to 3 minutes until pale, foamy, and slightly thickened. Whisk in the vanilla and almond extracts.
In a medium bowl, whisk together the almond flour, baking powder, and salt for about 30 seconds, breaking up any clumps with the back of a spoon.
Pour the orange puree into the egg mixture and whisk until the color is even. Add the almond flour mixture in two additions, folding gently with a spatula after each, until no dry streaks remain.
Pour the batter into the prepared pan, smooth the surface, and scatter the sliced almonds evenly over the top.
Bake for 30 minutes, then loosely tent the top with aluminum foil. Bake for another 25 to 35 minutes, for a total of 55 to 65 minutes, until the top is set and deep golden and a toothpick comes out clean.
Cool the cake in the pan on a wire rack for 30 minutes. Run a thin knife around the edge, release the springform ring, and let the cake cool completely for 1 to 2 hours.
Dust the top with powdered sugar through a fine mesh sieve, slide the cake onto a serving plate, and cut into 12 slices with a sharp knife, wiping the blade clean between cuts.