Preheat oven to 375°F. Cook sausage in a large skillet over medium-high, stirring often and breaking up meat using a wooden spoon, until cooked through, 6 to 7 minutes. Using a slotted spoon, transfer sausage to a 13- x 9-inch baking dish. Do not wipe skillet clean. Add butter to drippings in skillet, and melt over medium. Add bagel pieces; cook, tossing often, until bagel pieces are coated and lightly toasted, about 3 minutes. Add bagel pieces to sausage in baking dish. Set aside.

Place cream cheese in a large bowl, and slowly whisk in milk until combined. Add eggs, hot sauce, salt, pepper, white and light green parts of scallions, and 1 1/2 cups of the cheese; whisk until combined. Pour over bagel mixture in baking dish; toss lightly to evenly distribute all ingredients. Sprinkle evenly with remaining 1/2 cup cheese. Let casserole stand at room temperature 30 minutes to allow bread to absorb the egg mixture.

Bake uncovered, in preheated oven until eggs are set and top of casserole is golden brown, about 30 minutes. Sprinkle casserole with everything bagel seasoning and dark green parts of scallions. Let stand 10 minutes. Serve.