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In a saucepan over medium heat, melt 4 tablespoons of butter.
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Stir in the flour to create a roux. Cook for 1-2 minutes until it turns a light golden color.
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Gradually whisk in the milk and heavy cream to avoid lumps. Continue to whisk until the mixture thickens, about 5-7 minutes.
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Add salt, black pepper, and paprika. Stir to combine.
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Reduce heat to low and add the shredded cheddar, Gruyère, Fontina, and Parmesan cheeses. Stir until the cheese is completely melted and the sauce is smooth.