Preheat the oven to 350 degrees F. Lightly butter a 3-quart baking dish.
Arrange the bread cubes on a rimmed baking pan in a single layer. Bake until light golden, about 15 minutes. Transfer to a large bowl.
Melt 1 tablespoon of the butter in a large skillet over medium-high heat. Add the sausage and cook, breaking up any clumps, until brown, about 6 minutes. Transfer the sausage to the bowl with the bread using a slotted spoon. Reduce the heat to medium, add the remaining 3 tablespoons butter with the celery, fennel, onion, 3/4 teaspoon salt and pepper to taste and cook, stirring occasionally, until soft, about 8 minutes. Add the garlic and cook until fragrant, about 2 more minutes. Transfer to the bowl.
Add the broth, sage and thyme to the bowl and toss to coat thoroughly. If the mixture is very hot, cool until it is just warm, then add the eggs and toss to coat.
Increase the oven temperature to 400 degrees F. Transfer the stuffing to the prepared baking dish and bake, uncovered, until hot throughout and lightly browned on top, about 35 minutes.