Place eggs in a saucepan and cover with cold water by about 1 inch (2.5 cm). Bring just to a boil, cover and remove from the heat. Let large eggs stand about 12 minutes; size, starting temperature and elevation can require more time. Transfer to an ice-water bath for 10 minutes, then peel. Both whites and yolks must be fully firm: check an egg if uncertain and cook longer if needed. Wash hands after touching raw shells.

While the eggs cool, melt butter in a 12-inch skillet over medium-low heat. Add onion and cook 7–8 minutes, stirring, until soft and translucent. Reduce heat if the butter begins browning.

Add garlic and stir about 30 seconds. Add Cajun seasoning and paprika and stir about 20 seconds without scorching. Pour in the stock and simmer gently 4–5 minutes until the onions are tender and the sauce remains spoonable. Stir in Dijon.

Add the peeled whole eggs and turn gently in the sauce over low heat for 3–5 minutes, until hot. Add a splash of stock or water if the sauce tightens. Remove from the heat, stir in lemon juice and parsley, and taste before adding optional salt. Halve eggs just before serving if desired. Divide into 3 portions, each with 2 eggs and one third of the sauce. Optional bread, rice or potatoes are excluded from the estimates.

Refrigerate leftovers in shallow covered containers at 40°F (4°C) or below within 2 hours, or 1 hour above 90°F (32°C). Use within 3 days. Reheat thoroughly to 165°F (74°C), adding a little liquid and stirring gently to prevent scorching.