Set up an ice water bath: large bowl, lots of ice, a little water. Place a strainer or colander in sink and bring a pot of water to a boil, 4 inches deep.
Using a paring knife, remove the core from the top of each tomato and score the bottom of the skin of the tomato with an X. When water boils, gently place tomatoes in water and let them roll around 1 full minute. Transfer the tomatoes to the ice bath using a spider or tongs. Cool and peel tomatoes, letting them drain a bit as you work in the strainer. Slice and chop the tomatoes and place in a bowl.
Heat a soup pot over low heat with 2 tablespoons EVOO, 2 turns of the pan. Add the butter, and when it melts into oil, add Vidalia onions and season with salt and pepper. Do not let the onions brown. Let them sweat 10 to 15 minutes and add 1 cup water to onions, raise heat to medium-high, and let the water absorb and cook the onions to absolute mush. Add tomatoes, basil and stock and cook at medium boil 20 minutes to break down tomatoes. Remove basil stem and puree the soup using an immersion blender or transfer to food processor or high-power blender and puree, then return to pot. Simmer gently over lowest heat until ready to serve. Top with a few small leaves of torn basil.
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