How to Make It
Preheat oven to 350°F. Lightly butter a 9-by-13-inch or other shallow 3-quart baking dish; butter one side of a piece of foil, large enough to cover dish.
Heat olive oil in a large skillet over medium high. Add onions, chard stems, and ½ teaspoon salt and cook, stirring occasionally, until tender and beginning to turn golden, 6 to 8 minutes. Add chard leaves and ½ teaspoon salt, ¼ teaspoon pepper and cook, tossing, until wilted, about 3 minutes more.
Arrange bread and butternut squash slightly upright in prepared dish, alternating layers and shingling as you go. Transfer chard and onion mixture to dish, distributing evenly over bread and squash.
Whisk milk, eggs, and remaining ½ teaspoon salt and pepper in a large bowl until evenly incorporated. Pour egg mixture over baking dish, pressing everything gently to help soak up the liquid. Top with cheese, sprinkling a little in between the layers as well.
Cover with foil, buttered side down, and bake until slightly puffed, 25 to 30 minutes. Uncover and continue to bake until bread is golden brown, 20 to 25 minutes more. Let cool slightly before serving.