Preheat the oven to 350 degrees F. Spray a 3 1/2-quart casserole dish with nonstick spray.
Melt the butter in a large skillet over medium-high heat. Add the celery and onions and saute until beginning to soften, about 5 minutes. Stir in the sage, thyme and garlic and sprinkle with salt and pepper. Saute until very fragrant and all the vegetables are very soft and the onions are translucent, another 3 minutes. Stir in the chopped pecans in the last minute of cooking. Remove from the heat.
Add the cornbread and country bread to a very large bowl. Add the cooked vegetables and toss together.
Whisk together the broth and eggs in a medium bowl and sprinkle with salt and pepper. Pour the mixture over the bread and mix together with a wooden spoon until completely combined.
Pour the dressing into the casserole dish and bake until cooked through and golden on top, about 45 minutes.