Preheat the oven to 375 F.
Core the tomatoes and slice them in half, top to bottom. Remove the seeds with a spoon (or a quick squeeze over a bowl, and arrange the tomatoes cut side up on a baking sheet.
Sprinkle the tomatoes with 1 Tbsp of salt and 1 tsp of pepper.
Roast the tomatoes for about 45 minutes, until the tomatoes are very soft and have started to dry out a bit. Peel the roasted tomatoes and coarsely chop.
Add the olive oil to a pot, as well as onion, garlic, and remaining salt and pepper. Cook over medium heat until fragrant but not browned, about 8 to 10 minutes.
Add white wine and simmer for about 10 minutes, until most the wine evaporates. Add in tomatoes, stock and basil.
Cook for 20 minutes, until thoroughly heated.
While the soup cooks, pre-heat your pressure canner.
Process the soup in a blender, or use an immersion blender to make the soup completely smooth. If using a blender, you'll probably need to work in batches.
Once the soup is smooth, return it to the pot and bring it back to a simmer to prepare for pressure canning.
To can tomato soup, ladle the hot soup into prepared canning jars, leaving 1-inch headspace.
Add 2-piece canning lids and tighten to finger tight. Load into your preheated pressure canner.
Close the lid and vent the steam for 10 minutes, then add the canning weight and bring the canner up to pressure.
Once at pressure, process pint jars for 50 minutes and quart jars for 60 minutes, adjusting pressure to altitude.