In a large, heavy fry pan over medium heat, warm 3 Tbs. of the olive oil. Add the onions, cover and cook, stirring occasionally, until they are completely limp, 10 to 15 minutes. Reduce the heat to medium and cook, stirring frequently, until the onions are golden brown, 30 minutes. Remove from the heat and let cool. You will need 2/3 cup caramelized onions for the strata; cover and refrigerate the remaining onions.
Butter a 10-inch nonstick braiser. Put the bread in a large bowl. In a large sauté pan over medium heat, warm the remaining 2 Tbs. oil. Add the sausage and cook, stirring occasionally, until browned and cooked through, about 5 minutes. Using a slotted spoon, transfer to the bowl with the bread.
Return the pan to medium heat. Add the garlic and cook, stirring frequently, until fragrant, about 1 minute. Add the chard and cook, stirring occasionally, until wilted, 2 to 3 minutes. Transfer to the bowl with the bread mixture.
In another large bowl, whisk together the eggs and half-and-half. Pour the egg mixture over the bread mixture. Add the onions, cheese, salt and pepper and stir until well blended. Transfer the mixture to the prepared pan, cover with plastic wrap and refrigerate for 4 to 24 hours.
Preheat an oven to 350°F.
Remove the plastic wrap from the pan. Bake the strata until golden brown and cooked through, about 55 minutes. Let stand for 10 minutes before serving.