Canned whole peeled: pour into a large bowl with juices and crush by hand until roughly puréed — better texture than chopping. Fresh: core and roughly chop. Prep all aromatics (onion, garlic, carrot, celery) finely diced or minced before heating — the base cooks fast. San Marzano-style plum tomatoes preferred for lower acidity. Keep a wooden spoon or heatproof spatula close; sauce sticks and burns easily and needs regular stirring.

Heat olive oil in a heavy-bottomed saucepan over medium or medium-low. Add onion with a pinch of salt; cook, stirring occasionally, until soft and translucent, 10–15 minutes. Add garlic, stir constantly, 1–2 minutes. Add carrot or celery if using; cook until soft, 6–8 minutes. Stir in tomato paste if using, then add crushed or chopped tomatoes with their juices. Raise heat to a boil, then reduce to low for a steady simmer — small occasional bubbles only. Simmer uncovered, stirring often, 20–45 minutes: 20 min for cohesive sauce, 30 min for noticeable thickness, 45 min or more for deep reduction. Stir frequently — sauce burns easily as it thickens. Add a splash of water if it reduces too fast.

Sauce is done when thickened to your liking, raw sharpness has mellowed, and flavors taste rounded. Drag a spoon across the pan bottom — it should leave a brief trail before the sauce flows back. Taste and adjust salt, pepper, and a pinch of sugar if too acidic. Blend or pass through a food mill for smooth; leave as-is for chunky. Stir frequently in the final 10–15 minutes to prevent scorching. Use immediately or refrigerate in an airtight container for several days, or freeze for several months.