Preheat the oven to 400 degrees . In a large skillet, cook the bacon over medium heat until crisp. Add the onion, garlic and butter and cook until the onion is softened, about 5 minutes. Add the Swiss chard and cook until wilted, 3 minutes. Transfer to a colander and squeeze out any excess liquid. Reserve the skillet.
Add the cream and nutmeg to the skillet and boil until reduced by half, about 5 minutes. Let cool.
Stir the chard mixture into the cream. Add the oysters and 1/2 cup parmesan; season with salt and pepper. Spoon the mixture into a greased 3-quart baking dish.
In a small bowl, combine the remaining 1/2 cup parmesan and the bread crumbs; sprinkle over the gratin. Bake until golden and bubbling around the edges, 10 to 15 minutes. Serve warm.