Combine oil and butter 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 but not browned, about 3 minutes.
Add tomatoes, onion halves, basil sprigs, and sugar. Bring to a simmer, reduce heat to lowest setting (bubbles should barely be breaking the surface), and cook, stirring occasionally, until reduced by half, about 1 hour.
Discard onions and basil stems. Season to taste with salt. Allow to cool and store in covered container in the refrigerator for up to 2 weeks.