Position racks in the upper and lower thirds of the oven and heat the oven to 375°F.
Mix the bread cubes, oil, and 1 Tbs. of the garlic in a very large bowl, tossing to coat. Spread in a single layer on 2 large rimmed baking sheets; set the bowl aside.
Bake, stirring, swapping positions, and rotating the pans halfway through, until just golden, about 12 minutes. Return the bread to the bowl. (The bread can be prepared to this point up to 3 days ahead. Cool, cover, and store at room temperature.)
Lower the oven temperature to 350°F. Melt the butter in a 12-inch skillet over medium heat. Add the onions and celery and cook, stirring often, until tender, about 15 minutes. Add to the bread.
Put the skillet back over medium heat. Add the andouille and cook, stirring often, until browned, about 5 minutes.
Using a slotted spoon, transfer to the bowl of stuffing. Add the peppers and remaining 2 tsp. garlic to the fat in the pan and cook, stirring often, until just tender, about 6 minutes. Add to the stuffing along with the parsley, thyme, 2 tsp. salt, and 1 tsp. pepper. Season to taste with more salt and pepper.
Butter a 9x13-inch baking dish.
Whisk the eggs in a medium bowl to blend and then whisk in the broth. Pour the egg mixture over the stuffing and gently toss to combine. Let the stuffing sit until the liquid is absorbed, 5 to 10 minutes, tossing again if necessary. Add up to 1/2 cup additional broth if the stuffing seems dry; the mixture should be moist but not soggy.
Transfer to the prepared dish. Bake until lightly browned and crisp on top, about 40 minutes. Let stand for about 10 minutes before serving.