Preheat the oven to 250 degrees Fahrenheit. Line two large baking sheets with parchment paper.
Beat the egg whites in the clean bowl of a stand mixer fitted with the whisk attachment at medium high speed until soft peaks form. Add the sugars in a steady stream while the bowl is mixing, and allow it to continue to mix for several minutes until the meringue turns very smooth and shiny. Turn the mixer off an pinch some of the meringue between your fingers. If you can feel the granular sugar bits, mix it for a minute or two more, if you can’t, it’s ready. Add the vanilla, cardamom, and cinnamon and mix until combined.
Distribute the meringue into 4 roughly 8-inch flat discs, making two on each baking sheet. Place in the oven and bake for about 1 hour 15 minutes to 1 hour 30 minutes, or until the meringues are still on the outside when lightly tapped with the tip of your finger. Turn the oven off, leave the door slightly ajar, and allow it to cool to room temperature over the period of a couple hours (this will help keep it from cracking as much).
While that’s cooling, you can prepare the curd. In the top of a double boiler, whisk together the egg yolks and sugar for three minutes. Add the blood orange juice, Arbequina olive oil, and blood orange zest and continue whisking the mixture non-stop until the mixture thickens and turns a slightly lighter shade of orange, about 6 to 8 minutes. Immediately take it off of the heat and begin adding the butter to it, 1 tablespoon at a time, whisking constantly until each piece has melted before adding the next one. Cover with plastic wrap, making sure the wrap directly touches the surface of the curd to keep it from developing a film on top, and place it in the refrigerator.
Beat the cream in the bowl of an electric mixer with the whisk attachment at medium high speed until it begins to thicken. Gradually sprinkle in the sugar and vanilla. Continue mixing until it can hold a soft peak. Turn off, cover, and refrigerate.
Preheat the oven to 420 degrees Fahrenheit. Lay the slices flat on a sheet of parchment paper on a lipped baking sheet. Sprinkle them with the sugar. Place in the oven and bake until the edges of the oranges are golden and the sugar has melted, about 10 to 15 minutes. Remove and allow to cool to room temperature.To assemble the meringue stack, place a meringue on the serving plate. Spread about 1/3 of the whipped cream mixture over it, and spread about 1/4 of the curd mixture over that. Place about 1/3 of the broiled blood orange slices over the curd, and drizzle it with 1 tablespoon of the Arbequina olive oil. Repeat this two more times, so that you now have three layers stacked and the third layer has all the goodies on top of it. Place the final fourth meringue on top, and smear it with a dollop of the curd in the center. Drizzle the remaining tablespoon of the Arbequina olive oil over it, then place the two blood orange wedges on top. You can freeze it for an hour or two to make slicing easier, or just dive right in. Enjoy!