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preheat broiler.
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whisk eggs and milk in a medium bowl.
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mix in ½ cup cheddar; season with salt and pepper and set aside.
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heat oil in a large skillet, preferably cast-iron, over medium heat.
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add onion and sausage and cook, stirring occasionally, until onion is softened and sausage is brown, 6–8 minutes.
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add broccoli; season with salt and pepper and cook, stirring occasionally, until tender, 8 minutes longer.
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reduce heat to low and pour reserved egg mixture over vegetables.
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cook, shaking pan occasionally, until edges are just set, 10–12 minutes.
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top frittata with remaining ¼ cup cheddar.
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broil until top is golden brown and center is set, about 4 minutes longer.
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cut frittata into wedges and serve warm or room temperature.