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Heat olive in a large skillet over medium high heat. Add sausage and cook until browned, about 3 to 5 minutes, making sure to crumble the sausage as it cooks; drain excess fat, reserving 1 tbsp in the skillet.
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Add eggs to the skillet and cook, whisking; until they just begin to set. Gently whisk in half and half; season with salt and pepper, to taste. Continue cooking until thickened and no visible liquid egg remains, about 3 to 5 minutes; set aside.
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With a butter knife, spread butter on one side of each slice of bread. Place 2 slices, buttered side down, on work surface. Divide sausage, eggs and cheese evenly on both slices. Top with remaining slices, buttered side up.
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Heat skillet over low heat. Add sandwiches and cover; cook, turning once with a spatula, until golden brown and the cheese has just melted, about 2 to 3 minutes per side.
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Slice and serve immediately.