Place eggs in a saucepan in one layer and cover with cold water by 1 inch. Bring to a full boil, cover, remove from heat and let stand for 12 minutes. Both whites and yolks must be firm; allow extra time if necessary.

Transfer eggs to ice water for 10 minutes. Peel carefully, rinse off any shell fragments and pat dry. Keep whole or halve after warming.

Melt butter in a skillet over medium-low heat. Add onion and cook for 5–6 minutes until soft, stirring. Add garlic and cook for 30 seconds without browning.

Stir in paprika, oregano, optional cayenne, salt and pepper. Add the 2 tablespoons water and let the sauce gently bubble for 1 minute.

Add peeled eggs and turn gently in the sauce for 2–3 minutes until warm. Stir in lemon juice. If needed, add a teaspoon more water to loosen the sauce. Taste for salt, finish with optional parsley and serve immediately.

Refrigerate leftovers within 2 hours, or 1 hour above 90°F, at 40°F or below. Use within 3–4 days, respecting the eggs' original cooking date, and reheat to 165°F.