- Preheat oven to 250° and grease a baking dish. Scatter bread in a single layer on a rimmed baking sheet. Bake, stirring occasionally, until dried out, about 1 hour. Let cool; transfer to a very large bowl.
- Meanwhile, melt ¾ cup butter in a large skillet over medium-high heat; add onions and celery. Stir often until just beginning to brown, about 10 minutes. Add to bowl with bread; stir in herbs, salt, and pepper. Drizzle in 1¼ cups broth and toss gently. Let cool.
- Preheat oven to 350°. Whisk 1¼ cups broth and eggs in a small bowl. Add to bread mixture; fold gently until thoroughly combined. Transfer to prepared dish, cover with foil, and bake until an instant-read thermometer inserted into the center of dressing registers 160°, about 40 minutes.
- Continue to bake dressing, uncovered, until set and top is browned and crisp, 40–45 minutes longer.
Do ahead: Dressing can be baked (before browning) 1 day ahead. Uncover and let cool, then cover and chill. Uncover and bake until top is browned and crisp, 50–60 minutes.