Line a 9-by-13-inch baking dish with foil. In a large skillet, heat the butter over medium-high heat. Add the peppers and a pinch salt and cook, stirring, until the peppers are softened, about 10 minutes. Stir in the parsley and 3 tbsp. chives.
In a bowl, whisk together the eggs, 1/2 tsp salt and 1/4 tsp pepper. Whisk in the milk.
In the prepared dish, arrange 6 bread slices in a single layer, cutting to fit if needed. Top with the pepper mixture and 1 cup cheese, then the remaining bread. Pour in the egg mixture and top with the remaining cheese. [Make ahead: Cover with plastic; refrigerate overnight. Uncover before baking.]
Preheat the oven to 350 degrees . Bake until puffed up and browned, about 45 minutes. Let cool for 10 minutes. Sprinkle with the remaining 3 tbsp. chives.