Preheat the oven to 375°F. Use 1 tablespoon of the softened butter to grease a rimmed baking sheet; set the baking sheet aside and keep the remaining butter for the bread.
Spread the remaining 3 tablespoons of softened butter evenly on both sides of each of the 4 slices of bread.
Heat a skillet over medium heat. Working in batches if necessary, cook the buttered bread slices until golden brown and toasted on both sides (about 1–2 minutes per side). Transfer the toasted slices to the greased baking sheet in a single layer.
In a small saucepan over low heat, warm the 1 cup heavy cream until it just begins to simmer. Stir in the pinch of salt and the pinch of pepper, then remove the cream from the heat.
Drizzle about 1 to 2 tablespoons of the warmed cream evenly over each toasted slice of bread.
Crack one large egg onto the center of each slice of bread (one egg per slice), keeping the yolks intact. After the eggs are in place, pour the remaining cream over the eggs and bread, distributing it evenly.
Bake in the preheated oven until the egg whites are set and the yolks reach your desired doneness, about 13 to 15 minutes (13 minutes for mostly runny yolks; 15 minutes for firmer yolks).
Remove from the oven, sprinkle the 3 tablespoons chopped fresh chives evenly over the toasts, and serve immediately.