Prepare the Bread: Preheat the oven to 350°F. Use a 2-inch round cookie cutter or a knife to cut a circular hole in the center of 4 slices of toast. Set the cut slices and their circular cutouts aside for later use.
Make the Egg Custard: In a large mixing bowl, whisk together 4 of the eggs, the milk, salt, and hot sauce until smooth. Submerge the 4 slices of toast and their cutouts in the mixture, ensuring they are fully coated. Let the bread soak in the custard for about 10 minutes.
Cook the Sausage and Vegetables: Heat a 10-inch oven-safe skillet over medium-high heat. Add the olive oil and the sausage, breaking up the meat as it cooks. Sauté the sausage until it starts to brown, about 3 minutes. Stir in the sliced scallions and thyme, if using, and cook for another 2 minutes. Reduce the heat if necessary to prevent burning, then remove the skillet from the heat.
Assemble the Casserole Base: Arrange the uncut slices of toast over the cooked sausage mixture in the skillet, forming an even layer. Place the bread cutouts around the toast slices to fill in the gaps. Pour any remaining custard over the top to ensure everything is moistened. Bake in the preheated oven for about 15 minutes.
Add the Eggs and Cheese: Carefully crack one egg into each of the holes in the toast slices. Season the eggs lightly with black pepper. Sprinkle the grated Gruyère cheese evenly over the entire dish. Return the skillet to the oven and bake for another 15 minutes, or until the eggs are set to your liking.
Serve and Enjoy: Allow the casserole to cool slightly before serving. Serve warm, with an extra drizzle of hot sauce if desired.