Prep the potatoes: scrub and dice the potatoes into 1/2-inch (1.2 cm) cubes so they cook evenly and get maximum surface area for browning. Put the diced potatoes into a large pot, cover with cold water, add a generous pinch of salt, bring to a simmer, and cook 5–6 minutes until just tender but not falling apart. Par-cooking shortens skillet time and ensures a fluffy interior while allowing the exterior to crisp.

Drain and dry: drain the potatoes in a colander and spread them on a clean kitchen towel or baking sheet to steam-dry 5 minutes. Removing surface moisture is essential — wet potatoes will steam in the pan instead of browning, preventing that golden crust.

Brown the beef: while the potatoes dry, heat a large heavy skillet (10–12 inch) over medium-high heat until hot. Add 1 tbsp olive oil, then the ground beef, breaking it into pieces. Season with 1/2 tsp kosher salt and 1/4 tsp black pepper. Cook without stirring for the first 2 minutes to let a crust form (the Maillard reaction produces deep savory flavor). Then stir and brown until no pink remains, about 6–8 minutes total. If the beef releases a lot of fat, drain off all but about 1 tbsp to avoid a greasy hash.

Soften aromatics: push the browned beef to one side of the pan (or remove to a bowl if your pan is crowded). Add the remaining 1 tbsp olive oil to the empty side, reduce heat to medium, add the diced onion and the 3 minced garlic cloves, and cook 2–3 minutes until translucent and fragrant. Cooking aromatics in separate space and at moderate heat prevents garlic from burning and allows onion sugars to develop, adding sweetness to balance the beef.

Crisp the potatoes: add the par-cooked, dried potatoes to the skillet in a single layer if possible. Increase heat to medium-high. Let the potatoes sit undisturbed for 2–3 minutes to form a crust, then toss or flip to brown other sides. Continue cooking, stirring occasionally, until most cubes are golden and crisp — about 8–10 minutes. Keep the heat high enough for browning but low enough to avoid burning; the contrast between crisp exterior and tender interior is what defines a great hash.

Combine and season: stir the beef and softened aromatics back in (if you removed the beef earlier) and sprinkle the smoked paprika, remaining 1/2 tsp kosher salt (adjust to taste), remaining 1/4 tsp black pepper, and fresh thyme. Toss to combine and cook 1–2 minutes to marry flavors. At this point taste and correct seasoning — well-seasoned components make the final dish sing.

Cheese finish: lower the heat to low. Sprinkle the shredded cheddar evenly over the hash and cover the pan for 1–2 minutes until the cheese melts into pockets of creamy richness. Turning the heat down prevents the cheese from separating and gives you melty stretches rather than oily puddles.

Make garlic-infused oil for the eggs: while the cheese melts, use a separate 8-inch nonstick skillet. Add 1 tbsp butter and the remaining whole garlic clove smashed with the flat of a knife. Warm over low heat for 1 minute until fragrant — do not let the garlic brown. Remove and discard the garlic (or reserve for garnish) so the pan holds a garlic-scented butter without bitter, burnt pieces.

Fry the eggs: increase the heat under the egg skillet to medium-low. Crack two eggs into the pan (work in two batches if needed), season lightly with salt and pepper. Cook until the whites are set but the yolks are still runny, about 2–3 minutes. For a slightly set top, tilt the pan and spoon the warm garlic-butter over the whites (basting) — this gently cooks the top without flipping, producing glossy edges and tender yolks. Keeping yolks runny creates a natural sauce for the hash.

Assemble and serve: divide the cheesy beef & potato hash among four plates, top each portion with a garlic-fried egg, and finish with a sprinkle of chopped chives or parsley and a pinch of flaky sea salt. Serve immediately so the egg yolk mixes with the warm cheesy hash.

Storage note: refrigerate leftover hash in an airtight container for up to 3 days. Re-crisp in a skillet over medium-high heat, adding a splash of oil if necessary. Avoid microwaving if you want to preserve crispness.