Position racks in the upper and lower thirds of the oven; preheat to 250 degrees F. Spread out the bread on 2 baking sheets. Bake, rotating the pans and stirring halfway through, until lightly toasted, 1 to 1 1/2 hours.
Increase the oven temperature to 375 degrees F. Butter a 3-quart baking dish. Melt 4 tablespoons butter in a large pot or Dutch oven over medium-high heat. Add the ham, onion, celery, jalapeño, thyme, sage, 1 teaspoon salt and a few grinds of pepper; cook, stirring occasionally, until softened, about 6 minutes. Add the chicken broth and bring to a simmer, then remove from the heat. Let cool slightly.
Whisk the eggs in a large bowl. Stir in the bread and ham mixture. Transfer the stuffing to the prepared dish. Dot with 2 tablespoons butter. Cover with foil and bake 30 minutes.
Meanwhile, heat the remaining 1 tablespoon butter in a medium skillet over medium-high heat. Add the pineapple and cook until lightly browned, about 3 minutes. Add the brown sugar and stir until just dissolved, about 30 seconds.
Uncover the stuffing and scatter the pineapple on top. Bake until golden brown, 20 to 30 minutes.
Make the stuffing 2 to 3 days ahead; cover and refrigerate. Reheat at 325 degrees F, covered, until hot, then drizzle with broth or melted butter if it looks dry. Bake, uncovered, a few more minutes until crisp.