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Process mushrooms, onion, and garlic in a blender
until mixture is pureed. Melt 2 tablespoons butter
in a saucepan over medium heat. Cook and stir the
mushroom paste in the melted butter until smooth.
Melt 2 more tablespoons butter in a separate
saucepan over medium heat; gradually sprinkle
flour into the melted butter, stirring
continually. Whisk half-and-half into the flour
paste; cook and stir until the sauce is thick and
smooth, 4 to 5 minutes. Whisk mushroom mixture
into sauce and cook until sauce has thickened a
bit more, about 10 minutes. Stir cheese into the
sauce; cook until the cheese melts, 2 to 3
minutes. Season with salt and black pepper.
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