Adjust oven rack to middle position and preheat oven to 400 degrees. Coat a 9-inch by 13-inch baking dish with butter.
In large skillet over medium-high heat, cook pork until browned and cooked through, about 5 minutes. Transfer to bowl and wipe out pan with paper towels.
Over medium-high heat, melt butter until foaming. Stir in onion and celery and cook until translucent, about 7 to 8 minutes.
Meanwhile, whisk eggs in large bowl. Stir in broth, 1 teaspoon salt, and 1/2 teaspoon pepper.
To skillet, stir in parsley, sage, and thyme until fragrant, about 30 seconds. Transfer to bowl with eggs and toss to combine.
Add bread cubes, sausage, and cheeses and stir until uniformly combined. Pour into prepared baking dish.
Cover tightly with foil and bake 25 minutes. Remove foil and bake until crispy edges form, about 15 to 20 minutes longer.