Trim and discard ends of garlic cloves. Peel garlic and cut into slices about as thick as a quarter.
Pour tomatoes with all their juices into a large bowl and, using your (well-washed) hands, crush into small chunks. Throw away any hard stem ends.
Pour ¼ cup oil into the frying pan and place over medium heat. Add garlic and a couple of pinches of salt, and stir. Let garlic warm up and become fragrant—about 1 minute—but don’t let it brown. When you start to smell the garlic, pour in tomatoes with all the juice.
Using a wooden spoon, stir in remaining ¼ cup olive oil, 1 1/2 tsp. salt, the pepper, and basil sprigs. Turn heat to medium-high, bring to a simmer, cover, and cook 15 minutes. Turn off stove. Fish out and discard basil.
Use sauce as is, or, for a smooth texture, let it cool 5 minutes, then transfer carefully to a blender and purée on low for 30 seconds, holding down lid with a towel to prevent splattering.
MAKE AHEAD Up to 3 days, chilled airtight.