Once the water reaches a full boil, turn off the heat, cover the pan with a lid, and let the eggs sit in the hot water for 10–12 minutes, depending on how firm you like the yolks.
While the eggs rest, start the cream sauce. In a medium saucepan, melt the 4 tablespoons of salted butter over medium heat until just foamy but not browned.
Slowly pour in the heavy cream while stirring. Add the 1 teaspoon kosher salt. Bring the mixture to a gentle simmer over medium heat, stirring often so it doesn’t scorch on the bottom.
Let the cream and butter simmer gently, stirring frequently, until it thickens into a rich, velvety sauce that coats the back of a spoon, about 8–10 minutes. Adjust the heat as needed to keep it from boiling hard. Taste and add a pinch more salt if needed.
When the eggs are finished, drain off the hot water and immediately run cold tap water over them or place them in a bowl of ice water to stop the cooking. Let them cool until they’re comfortable to handle, 5–10 minutes.
Peel the eggs carefully, rinsing off any bits of shell. Pat them dry with a paper towel so they don’t water down the sauce.
Slice each egg in half lengthwise and arrange the halves, yolk side up, in the foil-lined, greased tray in a single snug layer.
Give the cream sauce one last stir. It should be thick, smooth, and steaming hot. If it has cooled or thickened too much, warm it gently over low heat with a splash more cream until pourable but still rich.
Pour the hot cream sauce evenly over the halved eggs in the tray, making sure every egg is well smothered. The sauce should pool slightly around them with a thick, glossy look.
Finish with a generous sprinkling of freshly ground black pepper over the top for that classic speckled look. Serve immediately while still bubbling and steaming, right from the foil-lined tray.
