Remove brown gills from the undersides of mushrooms, using a spoon; discard gills. Set aside.
Combine milk and flour, stirring with a whisk until blended; set aside.
Place a large nonstick skillet coated with cooking spray over medium-high heat until hot. Add carrots and next 3 ingredients; cook, stirring constantly, 5 minutes. Add spinach; cook, stirring constantly, 4 minutes or until spinach wilts.
Stir in flour mixture; bring to a boil. Reduce heat; simmer 1 to 2 minutes or until thick. Stir in cooked barley, 1/2 cup cheese, pepper, and nutmeg; cook 1 minute or until cheese melts.
Coat reserved mushroom caps with cooking spray; place on a broiler pan. Broil 5 minutes or until tender. Fill each cap with about 1/2 cup spinach-barley mixture, pressing firm to pack. Broil 3 1/2 minutes.
Combine breadcrumbs and remaining 1/2 cup cheese; sprinkle evenly on top of each mushroom. Broil 1 1/2 minutes or until cheese melts.