Preheat oven to 250°F. Butter a 13x9x2 baking dish and set aside. 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 3/4 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 1 1/4 cups broth and toss gently. Let cool.
Preheat oven to 350°F. Whisk 1 1/4 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°F, about 40 minutes. DO AHEAD: Dressing can be made 1 day ahead. Uncover; let cool. Cover; chill.
Bake dressing, uncovered, until set and top is browned and crisp, 40-45 minutes longer (if chilled, add 10-15 minutes).