Process tomatoes and their juice through food mill or pulse in food processor until pureed, or puree with hand blender.
Puree should not be completely smooth, and should have no chunks larger than 1/16-inch.
Set tomatoes aside.
Combine butter and oil in medium saucepan and heat over medium-low heat until butter is melted.
Add garlic, oregano, pepper flakes, and large pinch salt and cook, stirring frequently, until fragrant, being careful not to burn the garlic, about 30 seconds.
Add tomatoe mixture, basil, onion half, and sugar.
Bring to a simmer, reduce heat to lowest setting (bubbles should barely be breaking the surface), and cook, stirring occasionally fro about 1 hour.
Season to taste with salt and pepper.
Allow to cool and store in covered container in the refrigerator for up to 2 weeks.