1 tablespoon olive oil
1 pound andouille sausage links, quartered lengthwise, then cut crosswise into 1/4-inch-thick pieces (about 3 cups)
8 cups chopped onions (about 2 3/4 pounds), divided
5 cups chopped celery (about 10 stalks with leaves), divided
3 cups mixed chopped red, yellow, and/or green bell peppers, divided
1 teaspoon dried thyme
1/4 teaspoon cayenne pepper

2 large eggs
1/2 cup (or more) low-salt chicken broth
Green Onion-Jalape o Cornbread
1/4 cup (1/2 stick) butter, melted

Heat oil in heavy large pot over medium-high heat. Add sausage and saut until browned, about 6 minutes. Add 6 cups onions, 4 cups celery, and 2 cups bell peppers; saut until vegetables are brown, about 20 minutes. Add remaining onions, celery, and bell peppers, then thyme and cayenne; saut until vegetables begin to soften, about 5 minutes. Season with salt and pepper. (Can be made 1 day ahead. Cool slightly. Cover and chill.)

Preheat oven to 350F. Butter 15x10x2-inch glass baking dish. Whisk eggs and 1/2 cup broth in medium bowl. Crumble cornbread into very large bowl. Add vegetable and egg-broth mixtures; toss. Mix in more broth by 1/4 cupfuls if dressing is dry. Transfer dressing to prepared dish. Drizzle with butter.

Bake dressing uncovered until golden brown and slightly crisp on top, about 45 minutes.