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Preheat oven to 350 degrees.
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Heat a large skillet over medium heat. Add butter until melted and sizzling.
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Add diced potatoes to skillet, creating one flat layer over bottom of entire skillet.
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Cook for 7-10 minutes or until potatoes are crispy and evenly browned on bottom.
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Flip over as uniformly as possible and cook for another 7-10 minutes until other side gets nice and browned.
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Note on the potatoes: Do not over flip these as it may result in soggy potatoes! Give each side time to get nicely browned and crispy. You want to have the potatoes getting done about the exact time you want to mix everything together to throw into the oven. If potatoes are done cooking, and you are not quite ready for them, just flip again, let them continue to crisp and keep a close eye on them. If they are starting to get too dark, turn heat down to just keep them warm until further use.
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Meanwhile, heat a small saucepan over medium heat.
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Remove sausage link from casing and cook until browned, 5-7 minutes, breaking up into small pieces with spatula as it cooks.
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Put cooked sausage in large bowl to reserve for later.
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Return to saucepan and add beef broth and cooking wine.
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In a very small bowl, mix together about 1/4 cup of the broth and the cornstarch to prevent lumping. Add to sauce pan and whisk.
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Add dried rosemary.
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Continue heating broth until starting to bubble and thicken, about 5-7 minutes, whisking frequently to avoid lumps and sticking to the bottom of pan.
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Once gravy is thickened, add 1 cup of the Cheddar cheese and stir until melted. Remove from heat.
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Return to big bowl where sausage is reserved. Add basil leaves, roasted red bell pepper, and green onions. Add hot crispy diced hash browns straight from the pan. Quickly toss together until evenly distributed.
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Add gravy and stir together until evenly distributed. Mixture should have a decent amount of gravy, but not appear soupy. If you are concerned that you have too much sauce, reserve some of it to add after the initial mix.
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Place hash brown mixture in a 2.5 quart casserole dish.
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Top with remaining Cheddar cheese.
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Place in oven and cook for about 5 minutes, or until cheese is starting to melt.
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Turn to broil and cook for an additional 5 minutes, or until cheese starts to bubble and turns brown on top. (Watch closely to make sure it doesn't burn!)
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While cheese is melting, return to your large skillet.
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Add additional butter to coat pan (if needed). Pan should still be on medium heat and already preheated from before.
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Crack four eggs into pan.
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Fry until whites are set and yolks are still runny.
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Slide eggs on top of casserole and serve warm!