Cook bacon until crispy; crumble and set aside to cool with ham.
Whisk eggs, milk, thyme, chives, mustard, salt, and pepper in a large bowl until smooth.
Spray a 9x13-inch dish with cooking spray; arrange bread in bottom and scatter bacon, ham, and Gruyere over it.
Pour egg mixture evenly over bread and meat; press bread to submerge. Sprinkle with Parmesan, cover with plastic wrap, and refrigerate at least 8 hours or overnight.
Remove from refrigerator 30 minutes before baking; preheat oven to 350°F. Bake uncovered 45-50 minutes until puffed and golden brown.