Put the potato cubes in a 3-quart / 2.8 L saucepan, cover with cold water by about 2.5 cm / 1 inch, and add 1 tsp salt. Bring to a boil and cook for 6 to 7 minutes, until a knife slides in with a little resistance. Drain, then spread the potatoes on a clean towel for 2 minutes so they dry.
Heat 2 tbsp olive oil in a 30 cm / 12-inch skillet over medium-high heat until it shimmers. Add the potatoes in one layer and cook for 8 to 10 minutes, turning every 2 minutes, until golden on most sides.
Add the remaining 1 tbsp oil, the onion and the bell pepper. Cook for 5 minutes, stirring often, until the onion is soft. Stir in the garlic, smoked paprika, black pepper and chili flakes and cook for 30 seconds, until fragrant.
Add the halloumi cubes and cook for 3 minutes, turning once or twice, until the edges are golden. Don't add the halloumi earlier, or it darkens and turns rubbery before the potatoes finish crisping.
Reduce the heat to medium-low. Use the back of a spoon to make 4 wells in the hash, then crack an egg into each one. Season the eggs with black pepper, cover the skillet and cook for 4 to 5 minutes, until the whites are set and the yolks still jiggle.
Taste the hash away from the eggs and add a pinch of salt only if needed, since the halloumi is already salty. Scatter the parsley over the top and serve straight from the skillet within 5 minutes, with lemon wedges.