Heat oven to 375°F. Coat a 2-quart oval baking dish with nonstick spray.
Heat oil in a large nonstick skillet over medium heat. Add onion and cook 5 minutes. Add broccoli and cook another 3 minutes.
Place bread cubes in a large bowl. Stir in broccoli-onion mixture and the ham.
In a medium bowl, whisk together the eggs, milk, chives, mustard, salt and pepper. Pour over bread mixture, stirring to mix. Transfer to prepared dish, and top evenly with cheese.
Bake at 375°F for 40 to 45 minutes, or until strata registers 160°F on an instant-read thermometer.