Preheat oven to 350F. Arrange bread in a single layer on two rimmed baking sheets. Bake until toasted but not browned, about 10 minutes, rotating sheets halfway through.

Meanwhile, in a large skillet over medium-high heat, melt butter and cook onions, celery, and garlic until vegetables are softened, 5 to 7 minutes. Transfer to a large bowl. Stir in thyme, rosemary, salt, pepper, eggs, and toasted bread cubes. Start by adding 1 ½ cups of broth, and more until stuffing is moistened but not wet (there should not be any liquid in the bottom of the bowl).

Spoon stuffing into a buttered baking dish and cover with aluminum foil. Bake for 25 minutes, uncover and bake until golden, about 15 minutes more. Serve immediately.