Heat one tablespoon avocado oil in a cast iron on oven-safe skillet over medium low-heat. Slice the red onions very thinly and then add to the skillet. Saute on medium low heat until the onions caramelize and become sweet and light brown. As the onions are just about finished, add the dried marjoram and stir it in to cook through and heat. In total, onions will take about 30-40 minutes.
Meanwhile, cook the sweet potatoes. Heat oven to 400 degrees. Peel and slice the sweet potatoes into thin wedges. Drizzle 1-2 tablespoons avocado oil on a rimmed baking sheet, add the potatoes and toss until the potatoes are evenly coated with oil. Spread them out evenly and sprinkle with a good amount of salt and fresh ground black pepper. Roast the potatoes until they soft and start to brown - at least 20 minutes but this will depend on how thick the potatoes are. Check after 20 minutes and cook for addition 5-15 minutes, depending.
When the potatoes are done, add potatoes to skillet with the onions. Stir them gently to redistribute evenly. They should overlap and fill skillet.
Preheat the broiler.
In a large bowl, beat eggs and add a 1/4 teaspoon salt, freshly ground black pepper and cayenne. Pour egg mixture over the potatoes and onions. Gently lift the vegetables several times so the eggs can snuggle down in between. Remove the leaves from the marjoram sprig and sprinkle over top of the eggs. You can cover the pan at this point to get the eggs cooked evenly mostly the through and completely set on the bottom of the pan. Don't stir the eggs.
When eggs are mostly set, except for the top, place the skillet under the broiler. Watch carefully, the top of the eggs will simmer slightly, become solid (no liquid left on top) and brown ever so little. Remove from the broiler and allow to sit for 10 minutes of so before serving.