Preheat your oven to 375 degrees. Grease a 9x13 inch baking dish with butter or cooking spray and set it aside.

Crumble the day-old cornbread into a large mixing bowl. It should feel dry and slightly stale, not soft. If it feels at all moist, spread it on a baking sheet and put it in the oven for 10 minutes while the oven preheats to dry it out a little more. This step matters because too-wet cornbread makes the dressing dense and soggy instead of light and set.

Melt the butter in a large skillet over medium heat. Add the onion and celery and cook, stirring occasionally, for 7 to 8 minutes until softened and the onion looks translucent. Add the jalapenos and garlic and cook for another 2 minutes, stirring often, until the garlic smells fragrant. Remove the pan from the heat and stir in the thyme, salt, and pepper.

Scrape the sauteed vegetables into the bowl with the crumbled cornbread. Add 1 1/2 cups of the shredded cheddar, the beaten eggs, and 2 cups of the chicken broth. Stir everything together until fully combined. The mixture should look moist and hold together when you press it with a spoon. If it still looks dry, add the remaining 1/2 cup of broth a splash at a time. You want it to be moist but not soupy.

Transfer the mixture to the prepared baking dish and spread it into an even layer. Sprinkle the remaining 1/2 cup of cheddar over the top in an even layer.

Bake for 35 to 40 minutes until the top is golden brown and the center is set. To check, press gently on the center with a spoon. It should feel firm, not jiggly. The edges will pull away slightly from the sides of the pan when it is done.

Let the dressing rest for 5 minutes before scooping and serving. This helps it hold together better when you spoon it out.