Preheat oven to 400°.
Bring water to a boil in a large Dutch oven over medium-high heat. Add broccoli, and cook 5 minutes or until crisp-tender. Transfer broccoli to a large bowl with a slotted spoon. Add chicken to boiling water; reduce heat, and simmer 15 minutes or until done. Transfer chicken to a cutting board; cool slightly. Cut chicken into bite-sized pieces, and add chicken to bowl with broccoli.
Combine evaporated milk, flour, salt, pepper, and nutmeg in a saucepan, stirring with a whisk until smooth. Bring to a boil over medium-high heat; cook 1 minute, stirring constantly. Remove from heat. Add mayonnaise, next 4 ingredients (through soup), and 1/2 cup cheese, stirring until well combined. Add mayonnaise mixture to broccoli mixture; stir gently until combined.
Spoon mixture into a 13 x 9-inch baking dish coated with cooking spray. Sprinkle with remaining 1/2 cup cheese. Bake at 400° for 50 minutes or until mixture bubbles at the edges and cheese begins to brown. Remove from oven; let cool on a wire rack 5 minutes.