Prep the onions and garlic. Slice the onions thinly and mince the garlic. Thin slices caramelize more evenly and quickly; minced garlic will release flavor fast and should be added later so it doesn't burn.

Heat a large (10–12 inch) heavy skillet over medium heat and add 1 tablespoon olive oil plus 1 tablespoon butter. Use a heavy pan (cast iron or stainless steel) to promote even browning and heat retention. The butter adds a rounded flavor and helps the onions develop a deep golden color.

Add the sliced onions with a pinch (1/4 tsp) of the salt. Cook over medium heat, stirring every 2–3 minutes, for 12–15 minutes until deeply golden and soft. Lower heat if the edges start to burn. Slow, steady heat draws out the onions’ sugars and produces the sweet, jammy flavor that balances the beef.

When onions are starting to color, add the minced garlic and cook 45–60 seconds until fragrant. Garlic browns much faster than onions; adding it near the end prevents bitterness while still infusing the dish.

Push the onions to one side of the pan, increase heat to medium-high, and add the remaining 1 tablespoon olive oil. Add the ground beef in an even layer. Let it sit, undisturbed, for 2–3 minutes to develop a deep sear. Browning creates Maillard compounds—those savory, roasted flavors that make this dish feel restaurant-quality.

Break the beef into large chunks and continue cooking, stirring occasionally, until the beef is well-browned and no longer pink, about 4–6 minutes. Season with smoked paprika, the remaining 1 1/4 tsp salt and 1/2 tsp black pepper while the meat cooks so the spices bloom in the hot fat and adhere to the beef.

Stir the caramelized onions back into the browned beef so the juices and onion jam meld with the meat. Taste and adjust seasoning—if it needs brightness, add a small pinch of kosher salt or grind more pepper. The combination of onion fond and beef juices is the flavor base here.

Wipe out a small spot in the pan (or use a second small nonstick skillet) and add 1 tablespoon butter. Heat over medium-low. Crack eggs gently into a small bowl then slide them into the pan to cook sunny-side up. Cook slowly over medium-low so whites set without overcooking yolks—about 3–4 minutes. The butter fosters a tender edge and shiny yolk.

Season the eggs with a pinch of salt and the remaining 1/2 tsp black pepper. If you prefer the tops set but yolks runny, cover the skillet for 30–45 seconds. Placing eggs on top of the beef lets the residual heat finish them gently while the yolk acts as a sauce when broken.

Serve the beef and onions in shallow bowls or family-style from the skillet, nest an egg on top of each portion, and garnish with chopped parsley. The hot beef will mingle with the egg yolk, creating a luxurious, silky sauce—serve immediately.