Preheat the oven to 425°F (220°C). Spread the cubed potatoes on a rimmed baking sheet in a single layer; drizzle with 1 tablespoon olive oil, sprinkle with 1 teaspoon kosher salt, smoked paprika, and dried thyme, then toss to coat so each cube has a light film of oil and seasoning. Roast for 25–30 minutes, turning once halfway through, until golden brown and edges are irregularly crisp.

While the potatoes roast, bring a medium saucepan of water to a gentle boil for the eggs. Carefully lower the eggs into the water using a spoon. Cook for 6 minutes for soft but slightly set whites and runny yolks; adjust by ±30 seconds if you prefer firmer or softer yolks. Immediately transfer eggs to an ice bath for 3–4 minutes to stop cooking, then peel and set aside.

Pat the steak cubes dry with paper towels and season evenly with a pinch of kosher salt and 1/2 teaspoon coarse black pepper. Dry seasoning helps create a crisp sear.

When the potatoes have 5 minutes left to roast, heat a large heavy skillet over medium-high heat until very hot. Add 1 tablespoon olive oil and 1 tablespoon unsalted butter to the pan; when the butter foams and the oil shimmers, add the steak cubes in a single layer—do this in batches if necessary to avoid crowding. Sear for 1½–2 minutes per side, turning with tongs, until a deep brown crust forms but the interior remains medium rare to medium. Remove seared steak to a plate and let rest for 5 minutes; reserve pan juices.

In the same skillet with reserved juices, lower heat to medium and add the minced garlic. Sauté for 30–45 seconds until fragrant but not browned, scraping up any browned bits from the pan. Stir the garlic into the potatoes on the baking sheet or spoon the pan juices over the potatoes and steak to add gloss and flavor. If using lemon juice, add it now and stir lightly to deglaze and brighten the pan sauce.

Halve the peeled soft-boiled eggs lengthwise. Arrange bowls by placing roasted potatoes to one side, steak cubes on the other, and the egg halves in the center or offset—keep components separate so each is visible. Drizzle a little olive oil over the potatoes and steak for a finishing sheen and sprinkle coarsely cracked black pepper across the eggs and potatoes. Scatter chopped chives or green onions if using. Serve immediately while warm.

To store, allow components to cool slightly before transferring to containers. Keep eggs uncut and chilled separately from hot components to preserve yolk texture until reheating or serving. Reheat roasted potatoes and steak briefly in a hot skillet or oven to refresh crispness before plating.