Hardboil 5 large eggs, according to your preference. When finished, put them in an ice bath to stop their cooking. Meanwhile, prepare fry the capers the sauce.
Add enough olive oil to cover the bottom of a medium sized pot (you'll use it later for cooking the tomato sauce.) Heat the oil over medium-high heat and add the capers when the oil is hot. Stand back as the oil will spit and splatter. When the oil begins to settle down, after about 1 1/2 minutes, remove the capers from the pan with a slotted spoon and place them on a piece of paper towel.
Add the onion to the pot with the caper oil and saute on medium-high until it begins to soften and turn light brown, about 5 minutes. Add the garlic and tomato paste and cook until the tomato paste begins to caramelize, about 3 minutes. Add the diced tomatoes, red wine, black olives, balsamic, honey, and smoked paprika to the pan and bring it to a boil. Let it boil rapidly, stirring occasionally, until it thickens, about 10 minutes. Season to taste with sea salt and black pepper.
When the eggs are cool enough to handle, peel the shells and give them a quick rinse under running water to make sure no shell remains.
Cut the eggs in half lengthwise and scoop out the yolks and place them in a medium-sized bowl. Take the two least pretty egg whites, chop them fine then add them to the bowl with the yolks. Add the mayonnaise, olive oil, dijon mustard, white vinegar, salt, pepper, and cayenne and mix until well combined. Add half of the fried capers and stir through. Fill the egg whites with the egg yolks.
Preheat oven to 425 degrees.
Divide the tomato sauce between 4 ½-cup (4 oz) ovenproof dishes. Top each with 2 eggs and, if you're using, a sprinkle of parmesan cheese. Place them on a large baking sheet (to make it easier to move them) and bake in the oven for 10 minutes.
Serve with a dollop of yogurt, sliced green onions, black pepper, smoked paprika, and the remaining fried capers.