PREPARE boiling water canner. Heat jars and lids in simmering water until ready for use. Do not boil. Set bands aside.
WASH tomatoes; drain. Remove core and blossom ends. Cut into quarters. Set aside.
SAUTE onion and garlic in olive oil until transparent. Add tomatoes. Bring to a boil. Reduce heat and simmer 20 minutes, stirring occasionally.
PUREE tomato mixture in a food processor or blender, working in batches. Strain puree to remove seeds and peel.
COMBINE tomato puree and basil in large saucepot. Bring to a boil. Reduce heat and simmer until volume is reduced by half, stirring to prevent sticking.
ADD 1 Tbsp bottled lemon juice to each hot pint jar; Add 2 Tbsp. bottled lemon juice to each hot quart jar. Ladle hot sauce into hot jars leaving 1/2 inch headspace. Remove air bubbles. Wipe rim. Center hot lid on jar. Apply band and adjust until fit is fingertip tight. Place jar in boiling water canner. Repeat until all jars are filled.
PROCESS pint jars for 35 minutes and quart jars for 40 minutes. Remove canner lid. Wait 5 minutes, then remove jars and cool. check lids for seal after 24 hours; they should not flex up and down when center is pressed.