Gently toss together 2 tablespoons of the oil, the vinegar, thyme, garlic, tomatoes, red peppers and onions on a baking sheet. Sprinkle with salt and pepper. Arrange the tomatoes and red peppers skin-side up and bake until lightly charred, 45 to 50 minutes.
Warm the remaining 2 tablespoons of oil in a soup pot over medium-high heat. Stir in the tomato paste and mix well. Add the tomato and pepper mixture, including the juices, and combine. Stir in the stock; cover and bring to a boil. Reduce the heat to low and simmer, about 20 minutes.
Transfer the soup to a blender (or use an immersion blender) and pulse in small batches that reach only about one-third to halfway up the side of the jar. Be careful to start slow and then increase the speed to blitz the ingredients until smooth. Use a separate bowl to hold the blended soup, and then return it all to the pot.
Heat the soup to warm through, and then ladle into serving bowls. Garnish with the basil and serve with the Herbed Flatbread.
Roll the pizza dough out to about 1-inch thick. Sprinkle evenly with the herbs and roll into a log. Then knead 3 to 4 minutes to evenly incorporate the herbs. Reform the dough into a ball and place in a small bowl lightly coated with olive oil. Set aside to rest for 20 minutes.
Preheat the oven to 400 degrees F.
In a small saucepan over low heat, melt the butter with the garlic and red onions. Remove from the heat and keep warm. Brush a baking sheet with oil. Roll the dough out to about 1/8-inch thickness and transfer to the prepared baking sheet. Brush about half the garlic butter on top.
Remove the garlic and onions with a slotted spoon and scatter over the top. Bake until golden, about 25 minutes.
Remove from the oven, brush with the remaining garlic butter and sprinkle with salt. Cut into 2 inch strips and serve.
From Food Network Kitchens; after further testing and to ensure the best results this recipe has been altered from what was in the actual episode.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.