Saute mushrooms in 1 tablespoon of oil. Season with salt and set aside.
Whisk half-and-half, eggs, salt and pepper until smooth. Spray a 9-by-13-inch Pyrex or ceramic baking dish with vegetable cooking spray.
Line bottom with 6 slices of bread. Scatter half of the ham, and half of the mushrooms over the bread, then sprinkle with half of the scallions and cheese. Pour 1 cup of egg mixture over the top. Repeat layers with remaining bread, ham, mushrooms, cheese and scallions. Slowly pour remaining egg mixture evenly over top. Cover with plastic wrap, then weight down casserole with 3 16-ounce cans for at least 15 minutes to submerge ingredients. (Can be refrigerated overnight, but return to room temperature before baking.)
Adjust oven rack to middle position. Bake in preheated 325-degree oven until custard is just set, about 50 minutes. Turn on broiler and broil until strata is spotty brown and puffy (watch carefully), about 5 minutes longer. Let stand for 8 to 10 minutes, then serve immediately.