Grate cheddar and Parmesan and let them lose their refrigerator chill. If serving with pasta, reserve at least 1 cup hot pasta water.
Blend cottage cheese, 1/2 cup milk, cornstarch, Dijon, garlic powder, onion powder, salt and pepper for 60 to 90 seconds, scraping once, until glossy and completely curd-free.
Melt butter in a wide saucepan over medium-low. When it foams gently, reduce the burner to low without browning the butter.
Add the blended base and stir constantly for 3 to 5 minutes, until steamy and lightly coating the spatula. Keep below a simmer.
Remove from heat. Add cheddar in three handfuls, stirring each smooth before the next, then stir in Parmesan. Return to the lowest heat for only 10 to 15 seconds if necessary.
Add hot pasta water or warm milk 1 tablespoon at a time until slightly looser than the desired final consistency.
Taste, adjust salt and stir in optional lemon juice and herbs. Serve warm and do not boil the finished sauce.
