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In a large saucepan, melt the butter over medium heat.
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Once the butter is melted, add the flour and whisk continuously until the mixture turns golden brown and starts to smell nutty, about 2 minutes.
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Gradually pour in the milk, whisking constantly to prevent lumps from forming.
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Cook the sauce until it thickens enough to coat the back of a spoon, about 5-7 minutes.
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Reduce the heat to low and stir in the shredded cheddar and mozzarella cheese until melted and smooth.
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Season the cheese sauce with salt, pepper, and paprika if using. Taste and adjust seasoning if necessary.