Blanch the potato cubes for texture control: place the potato cubes in a pot, cover with cold water, add a generous pinch of salt, bring to a simmer and cook for 5 minutes — just until the exterior is tender but the center still firm. This removes excess starch and shortens skillet time so you can get crisp, cooked-through cubes without over-browning the beef.

Drain the potatoes and spread them on a tray or sheet pan to steam-dry for 5 minutes. Pat them dry with a kitchen towel — dry surface = better browning. While the potatoes steam-dry, mince the garlic and dice the onion; have the cheese measured and eggs at room temperature (room-temp eggs set more evenly and more quickly).

Heat a large (10–12 inch) cast-iron or heavy skillet over medium-high heat. Add 1 tablespoon olive oil and 1 tablespoon butter. The oil raises the butter's smoke point while the butter adds flavor — this combination gives both high-heat tolerance and a rich finish.

Add the parboiled, dried potato cubes in a single layer. Let them sear undisturbed for 3–4 minutes per side, turning occasionally, until many sides are deeply golden and crisp. Searing without constant stirring develops a crunchy crust and caramelization (Maillard reaction) that gives the dish textural contrast.

Once the potatoes have a good crust, push them to the perimeter of the pan and add the diced onion to the center. Cook the onion for 2–3 minutes until translucent and beginning to brown; then add the minced garlic and cook 30–45 seconds until fragrant. Cooking aromatics after the potatoes protects them from burning while ensuring they get time to sweeten and release aroma.

Move the potato and onion mixture to one side of the pan (or transfer to a bowl temporarily) and increase heat to medium-high. Add the raw ground beef to the open space, sprinkle with 1/2 tsp kosher salt and 1/4 tsp black pepper, and let it sear without stirring for 1–2 minutes to build a brown crust. Browning the beef separately creates flavorful fond (browned bits) that intensify the finished flavor.

Break the beef into bite-sized pieces, stir to brown all sides, then stir the beef together with the potatoes and onions. Add the smoked paprika and the remaining 1/2 tsp salt and 1/4 tsp pepper, plus the chopped thyme or rosemary if using. Stir to combine and cook another 2 minutes so the spices bloom in the hot fat — this brightens and deepens the seasoning.

Taste and adjust seasoning. Scatter the shredded cheddar evenly over the top of the beef-and-potato mixture; turn the heat to low. Cover the skillet for 1–2 minutes to help the cheese melt through the mixture—covering traps heat and steam so the cheese melts quickly without burning.

Using the back of a spoon, make four shallow wells in the hash. Crack an egg into each well. Season each egg lightly with a pinch of salt and a grind of pepper. Reducing the heat and creating wells protects the potatoes from scorching and lets the eggs cook gently on top of the hot hash.

Cover the skillet and cook on low for 4–6 minutes for runny yolks (cook longer if you prefer more set yolks). Alternatively, transfer the pan to a preheated 375°F (190°C) oven for 4–6 minutes. Keep a close eye — residual heat cooks quickly and you want that luscious runny yolk.

Remove from heat. Scatter a little extra cheese or chopped fresh herbs if desired. Serve hot straight from the skillet so each spoonful combines crisp potatoes, savory beef, gooey cheese and glossy runny yolk.