Combine first 3 ingredients. Spread 4 teaspoons cheese mixture evenly over each bread slice; set aside.
Heat oil in a large nonstick skillet over medium-high heat. Add mushrooms, and sauté 3 minutes. Sprinkle with 1/4 teaspoon salt and pepper. Remove mushrooms from skillet. Gradually add spinach to skillet; sauté 3 minutes or until spinach wilts, stirring frequently. Place spinach in a colander, pressing with the back of a spoon until barely moist. Coarsely chop spinach; stir into mushrooms.
Arrange 6 bread slices in a single layer in a 13 x 9-inch baking dish. Top with half of the spinach mixture. Repeat layer with remaining bread slices and spinach mixture. Combine 1/4 teaspoon salt, milk, eggs, egg white, and nutmeg; stir with a whisk until well-blended. Pour the milk mixture over the spinach mixture. Cover and chill for 8 hours or overnight.
Preheat oven to 400°.
Uncover strata, and sprinkle with breadcrumbs. Bake at 400° for 40 minutes or until set.