Cut bread into 1-inch cubes and let them sit out overnight, or toast in a 300°F oven for 10-15 minutes until slightly dry, but not browned.
In a large skillet, heat schmaltz over medium heat. Add onions and celery, cooking until softened (about 5 minutes). Add mushrooms, garlic, herbs, salt, and pepper. Cook until mushrooms release their moisture and begin to brown (another 6-8 minutes). Set aside.
In a large bowl, whisk together eggs, milk, cream, Dijon mustard, and a pinch of salt and pepper.
Grease a 9x13 baking dish with a bit of schmaltz. In a large bowl, toss toasted bread cubes with the mushroom mixture and half the cheese. Pour into the prepared baking dish. Slowly pour the custard over the top, pressing down lightly so the bread soaks it up. Let sit for 20-30 minutes to absorb, or refrigerate overnight if making ahead.
Preheat the oven to 375°F. Sprinkle remaining cheese on top and bake uncovered for 40-45 minutes, until the top is golden and the center is set. Let cool for 10 minutes before serving.
