Prep the chicken: Pat the chicken breasts very dry with paper towels (drier surface = better sear). If breasts are uneven, place each between two sheets of plastic wrap and lightly pound to an even thickness of about 1/2 inch (12 mm). This ensures uniform cooking so the thinner edges won't overcook while the center reaches temperature.

Season the chicken: In a small bowl, mix the salt, black pepper, smoked paprika and dried thyme. Sprinkle the seasoning evenly over both sides of the chicken. Let the seasoned chicken rest at room temperature for 8–10 minutes while you prepare the aromatics — this helps the seasoning penetrate and promotes an even cook.

Slice the onion thinly and smash (or finely mince) 6 of the 8 garlic cloves. Reserve two cloves whole and lightly crushed. Thinly slicing the onion lets it soften and caramelize evenly; smashing garlic releases its juices so it mellows and sweetens as it cooks.

Caramelize the onions: In a large stainless‑steel or heavy skillet over medium heat, add 1 tablespoon olive oil and 1 tablespoon butter. When the butter foams, add the sliced onions, a pinch of salt, and cook gently, stirring occasionally, for 10–12 minutes until deeply golden and sweet. Cooking the onions low and slow converts their natural sugars and produces a sweet, savory base that elevates the dish.

Add garlic and finish the confit: Push the onions to one side, add the remaining 2 tablespoons olive oil to the empty area, add the smashed garlic and the two lightly crushed whole cloves, and cook 1–2 minutes until fragrant and lightly browned — watch closely so garlic doesn't burn. Then stir the garlic into the onions and turn heat to low; keep the onion‑garlic mixture warm in the pan while you cook the chicken. These caramelized aromatics will become the sauce base and infuse the chicken with depth.

Sear the chicken: Heat a second large skillet (cast‑iron preferred) over medium‑high heat until very hot, then add a light film of oil (about 1 tbsp). Place the chicken breasts in the pan without crowding — you want an immediate sizzle. Sear undisturbed 3–4 minutes until a deep golden crust forms. That Maillard crust creates concentrated savory flavor and a pleasing texture contrast to the soft onion sauce.

Butter‑baste the chicken: Flip the breasts, reduce heat to medium, add 1 tablespoon butter and the reserved crushed whole garlic cloves to the pan, tilt the pan slightly and spoon the melted butter/garlic over each breast continuously for 2–3 minutes. Basting keeps the surface moist and transfers the butter’s nutty aroma into the meat. Cook until an instant‑read thermometer (inserted into the thickest part) reads 155–160°F (68–71°C) — carryover will raise the temperature a few degrees while the chicken rests.

Rest the chicken: Transfer breasts to a warm plate and loosely tent with foil for 6–8 minutes. Resting allows the juices to redistribute instead of running out when cut, keeping the meat juicy.

Create the pan sauce: While the chicken rests, place the onion‑garlic skillet back over medium heat and add 1/4 cup water to deglaze, scraping up any fond (browned bits). Simmer 1–2 minutes to reduce slightly, then whisk in the pan‑basted butter (scrape the chicken pan if you like to add extra flavor). The water loosens and dissolves the caramelized sugars and fond into a glossy sauce; finishing with butter enriches and slightly thickens it for a luxurious mouthfeel.

Fry the eggs: In a small nonstick pan over medium‑low, add a thin pat of butter. Crack each egg in gently; cook sunny‑side‑up until whites are set but yolks are still runny, about 3–4 minutes. Covering briefly for the last minute helps set whites without overcooking the yolk. The runny yolk becomes a rich finishing sauce when the dish is plated.

Plate: Spoon a generous bed of caramelized garlic‑onion sauce onto each plate, top with a butter‑basted chicken breast, then nestle a sunny‑side‑up egg on top. Spoon any remaining warm sauce and pan juices over the egg and chicken, and finish with a grind of black pepper. The egg yolk should break and mingle with the sweet onion sauce for a luxurious, saucy bite.

Finish and serve: Serve immediately while the chicken is warm and the egg yolk is runny. Suggest slices of crusty bread, roasted vegetables, or a simple green salad on the side to complete the meal.