Butter 9 by 13-inch baking dish
Line bottom snugly with a single layer of bread, cut to fit (edges can overlap slightly).  Make another layer of bread to fit on top, then set aside.

In a medium frying pan over medium-high heat, melt 2 tablespoons butter and add onion and mushrooms.  Stir often until liquid evaporates and mushrooms are lightly browned, about 15 minutes.  Spoon into a bowl.

Chop broccoli.  Melt remaining butter in pan and add broccoli; stir often until broccoli is very dry.  Stir in basil, pepper, and onion mixture.

Distribute half vegetable mixture over bread, then cover evenly with half the cheese.  Cover with reserved bread; repeat to make top layer.  Beat eggs and milk until blended and pour over casserole, moistening evenly.  Cover with plastic wrap and chill 8 to 36 hours.  Bake, uncovered, in a 350 degree oven until edges of casserole are lightly browned and center is firm when touched, about 45 minutes.  Cool for 15 minutes.  Cut into rectangles.  Serves 6 to 8.