For the Creole sauce: Whisk together the mayonnaise, mustard, Cajun seasoning, paprika, pickle, hot sauce, Worcestershire sauce, garlic and chives in a small bowl until combined. Cover and refrigerate until the skillet is done.
For the skillet: Preheat the broiler. Toss the shrimp with the House Seasoning in a small bowl and set aside.
Melt the butter in a large (12-inch) cast-iron skillet over medium heat. Add the onions and green and red peppers and cook, stirring occasionally, until the onions begin to turn translucent, about 5 minutes. Add the shrimp, andouille and garlic and cook, stirring constantly, until the shrimp are opaque and the sausage is heated through, about 2 minutes.
Add the eggs and milk and slowly scramble the mixture with a silicone spatula, scraping up large curds from the bottom of the pan. As soon as the eggs are almost set, but still a tiny bit runny (about 2 minutes), remove from the heat and sprinkle with the Cheddar.
Broil until the cheese melts, 10 to 15 seconds. Remove the skillet from the oven, drizzle with some Creole sauce, and sprinkle with the green onions.
Miss Brown’s House Seasoning:
Stir together the garlic powder, onion powder, paprika, salt and pepper in a small bowl.
Miss Brown’s House Seasoning can be made in a large batch and stored in an airtight container at room temperature for about 3 months.