Preheat oven to 350°F (175°C).

In a large bowl, combine cubed French bread, cubed ham, grated Cheddar cheese, and sliced mushrooms.

In a separate bowl, whisk together dry mustard, flour, and melted butter until smooth.

Gradually whisk in eggs and milk until well combined.

Pour the egg mixture over the bread mixture, ensuring all ingredients are evenly coated.

Place one-third of the mixture into a greased 2 1/2-quart casserole dish.

Bake for 30 minutes, or until golden brown and set.