Place a 10-inch cast-iron skillet in oven; preheat oven to 450°F. (Do not remove skillet.) Heat skillet 7 minutes.
Meanwhile, stir together the first 5 ingredients and 1 teaspoon of the salt in a large bowl. Whisk together buttermilk and 2 eggs in a medium bowl.
Remove skillet from oven, and add 2 tablespoons butter to skillet. Return to oven. Stir buttermilk mixture into cornmeal mixture until just combined. Stir in 1/2 cup of the melted butter. Pour mixture into hot buttered skillet; return to oven. Bake until golden brown and cornbread pulls away from sides of skillet, 18 to 20 minutes. Transfer skillet to wire rack; cool 15 minutes. Crumble cornbread into a large bowl. Set aside. Reduce oven heat to 400°F.
Cook bacon in a large skillet over medium-high until crispy, 8 to 10 minutes. Transfer to a plate lined with paper towels to drain; reserve 3 tablespoons drippings in skillet. Add celery and onion to skillet; cook, stirring occasionally, until tender, about 8 minutes. Stir in garlic; cook 30 seconds. Add kale, thyme, pepper, and remaining 2 teaspoons salt; cook, stirring often, until kale is tender-crisp, 8 to 10 minutes. Add 1/2 cup of the broth; cook, stirring, until evaporated.
Add kale mixture, bacon, and remaining 3 cups broth to cornbread mixture in bowl; gently stir to combine. Beat 3 eggs. Add beaten eggs and remaining 1/2 cup melted butter to cornbread mixture; gently stir to combine. Spoon mixture into a greased 13- x 9-inch baking dish. Bake at 400°F until golden brown, 35 to 40 minutes.