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.