Hard-cook the eggs: Place the eggs in a small saucepan and cover with cold water by about 1 inch. Bring to a gentle boil over medium-high heat, then cover, turn off the heat, and let sit for 10–12 minutes. Transfer the eggs to a bowl of ice water and cool completely.
Peel and chop the eggs: Once cool, peel the eggs and pat them dry. Roughly chop them into small pieces—about 1/4-inch chunks—so they’ll distribute nicely through the cream sauce. Set aside.
Make a simple roux: In a medium saucepan over medium heat, melt the butter. When it’s fully melted and just beginning to foam, sprinkle in the flour. Whisk constantly for 1–2 minutes, until the mixture is smooth and smells slightly nutty but has not browned deeply. This cooks out the raw flour taste.
Add the milk and thicken the sauce: Slowly pour in the milk while whisking continuously to avoid lumps. Continue to cook, whisking often, until the mixture comes to a gentle simmer and thickens to a creamy, sauce-like consistency that coats the back of a spoon, 4–6 minutes. If it becomes too thick, whisk in a splash more milk.
Season and add the eggs: Season the sauce with salt and pepper to taste. Gently stir in the chopped eggs, folding them through the sauce so they’re evenly distributed. Reduce the heat to low and warm the mixture for 1–2 minutes, stirring occasionally, just until the eggs are heated through. Avoid boiling, which can make the sauce grainy.
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