Heat butter and olive oil in a small saucepan over medium heat; cook and stir onion and garlic until tender and fragrant, about 1 minute. Add flour and whisk until flour is lightly browned and onion is translucent, 2 to 3 minutes.
Mix milk, Parmesan cheese, basil, oregano, salt, and black pepper into onion mixture. Cook, whisking continuously, until cheese is melted and sauce has thickened slightly, about 5 minutes. Remove from heat.
Mozzarella cheese or 3-cheese Italian cheese blend can be used in place of the Parmesan cheese, if desired.
Dried basil can be used in place of the fresh basil.
Add more milk if sauce is too thick in step 2.
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