1. Preheat oven to 375 degrees. Heat a large skillet over medium-high. Add sausage and cook, breaking up meat with a wooden spoon, until browned, 5 minutes. Transfer sausage to a small bowl, leaving behind as much fat as possible. Add shallots and chard stems to skillet and cook over medium until shallots are translucent, 3 minutes. Add a little water to pan and cook, stirring and scraping up browned bits from skillet with spoon. Add chard leaves and cook, stirring occasionally, until wilted, 3 minutes. Season with salt and pepper.

2. Spread half the sausage in a 9-by-13-inch baking dish and top with half the chard mixture. Top with bread, overlapping slices slightly. Top with remaining sausage and chard mixture. In a large bowl, whisk together milk and eggs, season with salt and pepper, and pour evenly over top. Firmly press on strata to submerge bread in egg mixture. Cover dish tightly with foil and let sit 10 minutes (or refrigerate up to overnight).

3. Bake strata 45 minutes. Uncover, sprinkle with cheese, and bake until cheese melts and egg mixture is mostly absorbed and set in center, 6 to 10 minutes. Let cool 10 minutes before serving.