Heat oven to 375° F. Butter a 9-by-13-inch baking dish. Divide the bread between 2 rimmed baking sheets and bake until dry and crisp, 10 to 12 minutes.
Meanwhile, melt the butter in a large skillet over medium heat. Add the onions, celery, 1 teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until very tender and beginning to brown, 10 to 12 minutes. Add the wine and cook until evaporated, 2 to 4 minutes; transfer to a large bowl and let cool for 10 minutes.
Add the bread, broth, eggs, figs, pine nuts, thyme, and ½ teaspoon salt to the vegetables and toss to combine. Transfer to the prepared baking dish. Cover with buttered foil and bake for 20 minutes. Uncover and bake until browned, 20 to 30 minutes more.
