Preheat broiler with rack 4 inches from heat. Heat oil in a large, ovenproof frying pan over high heat. Cook sausage, stirring often, until browned, about 3 minutes. Add bell pepper, mushrooms, onion and cook until vegetables are softened, about 3 minutes. Stir in swiss chard, reduce heat to medium and cook, stirring occasionally, until wilted, about 10 minutes.

Meanwhile, whisk eggs in a large bowl until starting to foam. Stir in cheese and salt. Pour mixture into fry pan with vegetables; cook on stove until the bottom sets, about 3 minutes. Broil until firm and browned, 3 minutes.