Heat oil in a large nonstick skillet over medium-high heat. Add shallots; sauté 3 minutes or until translucent. Add mushrooms, and sauté 7 minutes or until mushrooms release moisture. Cool slightly. Stir in ham and parsley.
Arrange half of bread in a 13 x 9-inch baking dish coated with cooking spray. Top bread with half of the mushroom mixture and 1/2 cup of shredded cheese. Repeat layers with remaining bread, mushroom mixture, and 1/2 cup cheese.
Combine milk and next 4 ingredients (through pepper), stirring with a whisk; pour evenly over bread mixture. Cover with foil coated with cooking spray. Refrigerate for 8 hours.
Preheat oven to 350º.
Remove strata from refrigerator, and let stand at room temperature 15 minutes. Bake strata, covered, at 350º for 30 minutes. Uncover and bake an additional 20 minutes or until set and golden brown. Sprinkle with chives, if desired.