Quarter your tomatoes into chunks.  If they're very large, you can chop them into eighths instead.   Romas sometimes don't need to be chopped at all.

Fill your blender with the chopped tomatoes, then blend until smooth.  If you prefer a chunkier sauce, you can blend in less time or skip the blending entirely (although I have never tried skipping it).

Blending turns the tomatoes into a foamy, pink puree.  Pour this into an 8-qt stock pot.  Continue quartering, blending, and pouring until all the tomatoes are in.

Turn the pot on medium heat and allow it to slowly come to a boil.  The foam will rise to the top, and after a while, it will look more like normal 'tomato sauce' (i.e., red liquid), but it will be really thin.  Continue to let it simmer for an hour or so until it's partially cooked down.

When the sauce is about half as thick as you'd like, chop the onion and garlic.  Add the veggies, salt, and basil to the pot.  (Your kitchen will now smell amazing.)

Continue to simmer the sauce on medium heat until it is as thick as you'd like.

Fill clean jars with hot sauce, leaving approximately 1" headspace (I fill to the bottom of the threads).

Lid the jars and place them in a large stock pot, properly set up to can.  Fill with hot water and turn it on.

Process quarts for 30 minutes (from a rolling boil).

Remove jars from the pot and set them on the counter to cool.  Watch for them to seal within the first 30 minutes or so.  If they don't seal within 24 hours, re-process or put them in the fridge to use right away.