Make the herb butter: Mix the butter with the sage, thyme, parsley, rosemary, tablespoons salt and 2 teaspoons pepper in a small bowl until well combined.
Make the stuffing: Melt 1/2 cup of the herb butter in a medium skillet over medium heat. Add the onions and celery and saute, stirring occasionally, until translucent, 10 to 15 minutes. Remove from the heat and let cool.
Preheat the oven to 375 degree F. Combine the cooked onions and celery with the cornbread, egg, chicken broth, milk and 3/4 teaspoon each salt and pepper in a large bowl; mix well. Place in a buttered 9-by-13-inch baking dish. Cover with foil and bake 30 minutes; remove the foil and bake until golden brown, about 20 more minutes.