Make this crustless quiche in a springform pan for a pretty (and sliceable) presentation. Cook the cheese grits and sausage up to two days in advance; bake the quiche the morning of the brunch.
Bring milk and butter to a simmer in a large saucepan over medium-high, stirring occasionally, 5 to 7 minutes. Stir in grits, 1/2 teaspoon salt, and 1/4 teaspoon pepper, and cook, stirring often, until grits are tender and mixture is thickened, about 15 minutes. Remove from heat, and stir in Cheddar and processed cheese until smooth. Pour grits mixture into a lightly greased 9-inch springform pan. Let cool to room temperature, about 30 minutes. Cover loosely with aluminum foil, and chill 8 to 24 hours.
Meanwhile, cook sausage in a large nonstick skillet over medium-high, breaking sausage into small pieces with a wooden spoon, until browned, 6 to 8 minutes. Transfer to a plate lined with paper towels to drain well. Let cool to room temperature, about 10 minutes. Transfer to an airtight container, and chill 8 to 24 hours.
Preheat oven to 325°F. Place springform pan on a rimmed baking sheet lined with aluminum foil. Whisk together eggs, cream, Dijon mustard, and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper in a large bowl. Fold in scallions and cooked sausage. Pour evenly over grits mixture in prepared pan. Bake in preheated oven until just set, about 1 hour and 10 minutes. Sprinkle with Parmesan cheese, and let stand 20 minutes.
Run a sharp knife around edges of pan, and remove outer ring. Sprinkle quiche with chives just before serving.