For fresh tomatoes, bring a large pot of water to a rolling boil and prepare a large bowl half-filled with ice cubes and cold water; cut a ¼-in (6-mm) deep X into the bottom of each tomato. Boil the tomatoes for 1 to 2 minutes, stopping when the skins wrinkle and begin to split but the tomatoes still feel firm.

Lift the tomatoes out with a slotted spoon and place them in the ice water for a few minutes, then peel away the loosened skins. The cold bath cools them enough to handle, and the skins should slip off without pulling away much flesh.

Remove the cores, cut the tomatoes into ½-in (13-mm) pieces, and put them with every bit of juice into a Dutch oven or saucepan. The tomato pieces will look loose and watery at this stage, which is exactly right before reduction begins.

Add the butter, olive oil, onion halves, smashed garlic, salt, and sugar, then bring the mixture just to a gentle boil. Lower the heat as soon as bubbles reach the surface so the sauce settles into a quiet, steady simmer.

Simmer uncovered for 1½ to 2½ hours, stirring occasionally and pressing the softening tomatoes against the pan with a wooden spoon or potato masher. The sauce is ready when watery bubbling has changed to glossy, thick tomato ribbons that mound briefly on the spoon.

Remove and discard the onions, mash any remaining large pieces of tomato and garlic, then taste and adjust the seasoning before stirring in the basil. The finished sauce should be slightly rustic yet spreadable, with a sweet, softened tomato aroma and basil that smells fresh.