Par-cook the potatoes: Microwave diced potatoes in a covered, microwave-safe bowl with a splash of water for 4–5 minutes, until just tender. Drain and pat dry. This jump-starts cooking and helps them crisp.

Heat the skillet: Place a large cast-iron or heavy skillet over medium-high heat. Add oil and butter. When shimmering, add the potatoes in an even layer. Don’t stir for 3–4 minutes to build a crust.

Season and crisp: Sprinkle with salt and pepper. Flip and cook another 4–6 minutes, stirring occasionally, until golden and crisp-edged. Transfer to a plate and keep warm.

Char the corn: In the same skillet, add a touch more oil if needed. Add corn in an even layer and cook undisturbed for 2–3 minutes to get some char. Stir and cook 2–3 minutes more.

Sauté aromatics: Reduce heat to medium. Add onion and jalapeño. Cook 3–4 minutes until softened. Stir in garlic, chili powder, and smoked paprika. Cook 30 seconds until fragrant.

Make the elote-style sauce: In a small bowl, mix mayonnaise, sour cream, lime zest, and 1–2 teaspoons lime juice. Season with a pinch of salt and pepper. Adjust lime to taste.

Combine the hash: Return potatoes to the skillet. Toss gently with the corn mixture and half of the elote sauce. Add half the cotija and half the cilantro. Taste and adjust salt, pepper, and lime.

Cook the eggs: Create 4 small wells in the hash. Crack an egg into each. Lower heat to medium-low, cover, and cook 4–6 minutes until whites set and yolks are done to your liking. For runnier yolks, check at 4 minutes; for jammy, go 5–6.

Finish and serve: Drizzle with remaining elote sauce. Top with the rest of the cotija and cilantro. Add green onions and hot sauce if you like. Serve straight from the skillet with lime wedges.