Begin by peeling and chopping the potatoes into even pieces. Place them in a pot of salted water and boil for 10-15 minutes until tender.

In a separate pot, bring water to a boil. Carefully add the eggs and boil for 8-10 minutes; after cooking, transfer the eggs to a bowl of cold water to stop the cooking process.

Slice the leeks (or green onions) and rinse them thoroughly to remove any dirt. Boil the leeks in salted water for about 4-5 minutes until tender, then drain them well.

While preparing the rest of the ingredients, preheat your oven to 375°F (190°C). Grease an oven-proof dish with the remaining butter to prevent the casserole from sticking.

Spoon the creamy leek and potato mixture into the greased casserole dish, spreading it evenly across the bottom.

In a saucepan over medium heat, melt the remaining butter, then sprinkle in the flour, whisking until a paste forms. Gradually pour in the warm milk, stirring continuously until the sauce thickens.

Stir in the nutmeg, adding a hint of warmth to the cheese sauce. Season with salt and pepper according to your taste. Fold in most of the sharp cheddar cheese.

Arrange the halved eggs cut-side up on top of the potato mixture in the casserole. Pour the thick cheese sauce over the eggs and potatoes, ensuring even coverage.

Place the casserole in the preheated oven and bake for 20-25 minutes until golden and bubbling.