Preheat oven to 350°F. Grease/prepare an 8x8 baking dish.
Melt butter in large skillet over medium-high heat; add onions and celery and cook until lightly browned, about 8-10 minutes.
Add to bowl with bread; stir in herbs, salt, and pepper. Add half broth and toss gently. Let cool.
Whisk remaining broth and egg in a small bowl. Add to bread mixture; fold gently until thoroughly combined.
Add to baking dish, cover with foil, and bake until until dressing is steaming, about 30 minutes.
This can be made a day or two in advance.