Preheat the oven to 375 degrees F; butter a shallow 3-quart baking dish. Melt 6 tablespoons butter in a large pot or Dutch oven over medium-high heat. Add the onions and celery and cook, stirring occasionally, until softened, about 5 minutes. Add the mustard greens and cook, stirring, until wilted, about 3 minutes. Stir in the ham, thyme and sage. Add the broth, 1 teaspoon salt and a few grinds of pepper; bring to a simmer, then remove from the heat.
Whisk the eggs in a large bowl. Add the bread cubes and ham-broth mixture and stir until combined. Transfer the stuffing to the prepared baking dish. Cut the remaining 2 tablespoons butter into small cubes and scatter evenly over the stuffing. Cover with foil and bake 30 minutes, then uncover and bake until golden, 20 to 30 more minutes.