Position a rack in lower third of the oven; preheat to 375 degrees . Grease a 3-qt. baking dish. In a blender, mix the cream cheese, milk and butter until smooth.
in a large bowl, combine the hash browns, cream cheese mixture and 3/4 cup scallions. Season with salt and pepper. Transfer to the baking dish; smooth top. Using a ladle, make six shallow indentations in the hash browns. Bake until the edges are brown, about 45 minutes.
Remove the casserole from the oven. Preheat the broiler. Crack an egg into each indentation and season. Broil until the whites are set, 3 to 4 minutes.
Place a slice of salmon and a dollop of sour cream next to each egg. Garnish with the lemon zest and scallions.