Spread bread cubes in a single layer on 2 large baking sheets. Bake at 350° for 10 minutes or until toasted. Let cool, and transfer to a very large bowl.
Melt 1 Tbsp. butter in a large deep skillet over medium-high heat; add country ham, and sauté 2 to 3 minutes. Add ham to bread in bowl.
Melt remaining butter in same skillet over medium heat; add onion and celery, and sauté 8 minutes. Add garlic, and sauté 2 minutes. Remove from heat; add to bread in bowl. Combine broth, egg, and pepper. Add broth mixture to bread, tossing well. Stir in fresh herbs. Spoon dressing into a lightly greased 13" x 9" baking dish.
Bake, covered, at 400° for 30 minutes. Uncover and bake 15 to 20 more minutes or until top is browned and crusty.
Make Ahead: Prepare dressing a day ahead. Store dressing, unbaked, in refrigerator. The next day, let dressing stand at room temperature 30 minutes and bake as directed. Reheat briefly just before serving.