Cut 1 inch from each end of onions. Stand onion on a flat side, put the point of a sharp knife in the very center, and make 1 cut, straight down from top of onion to bottom.

Bring a saucepan with 6 to 8 cups salted water to a simmer; add onions and simmer until they soften enough that layers can be easily separated, 20 to 30 minutes. Remove onions from water with a slotted spoon; let cool. Do not discard onion cooking water.

Meanwhile, heat olive oil in a skillet over medium-high heat and add half the beef; cook and stir, breaking up beef with a spatula until browned and crumbly, about 5 minutes. Add 1 cup diced onions, black pepper, salt, and cayenne, and cook, stirring, until onions turn translucent, about 5 minutes.

Add garlic, dried oregano, and tomato paste, and cook, stirring, for a few minutes. Add rice and stir until evenly coated. Add vinegar and chicken broth, lower heat to medium, and cook, stirring, until rice has absorbed all the liquid, about 20 minutes. Remove from heat, and let cool to room temperature, about 15 minutes.

Add remaining beef, feta, parsley, mint and oregano to a bowl, along with cooled rice mixture. Mix thoroughly to combine. Refrigerate until needed.

For tomato broth, measure 4 cups onion cooking water into a saucepan; stir in tomato sauce and chicken bouillon paste. Bring to a simmer over high heat, reduce heat to low, and keep hot until needed.

Preheat the oven to 375 degrees F (190 degrees C).

Separate onion sections and set aside the best 18 pieces, ideally using the ones from the middle of the bulb that are long enough to cover the filling. If desired, tear or chop extra onion, and add to the bottom of a large baking dish.

Add about 2 rounded tablespoons of filling to each onion section, and roll up to cover filling. Place stuffed onions in the baking dish, and ladle tomato broth over the dish. Cover with foil, and place on a sheet pan to catch drips.

Bake in the preheated oven for 1 hour. Remove foil and continue baking, uncovered, until onions are tender, and filling is fully cooked, about 1 hour more. If the dish appears too dry, you can place foil back on. Let rest 10 minutes before serving.