Grate cut faces of tomatoes with a box grater set over a large bowl. Compost skins.
Combine grated pulp, tomato paste, and garlic in a medium saucepan set over medium-high, and season with salt. Bring to a boil, then reduce to a simmer, and cook until sauce has reduced by half, about 10 minutes. Stir in basil sprig, then transfer sauce to jars. The sauce will keep for up to 5 days in the fridge, or up to 6 months in the freezer.