For the dressing: Place the shallots, olive oil and vegetable oil in a small saucepan over low heat. Cook the shallots gently, at barely a bubble, until completely soft, 15 minutes. Remove from the heat and let cool to room temperature.
To a food processor, add the cooked shallots and oil, and the cider vinegar, iced tea, mustard, honey and salt. Pulse until the shallots are finely chopped and the dressing is thick. Set aside.
For the salad: Bring a medium pot of salted water to a boil over medium-high heat. Add the fregola and cook for 8 minutes or until cooked through. Drain well, then place in a large bowl. While the fregola is still warm, add the dressing and toss to coat. Allow the fregola to cool slightly before adding the green beans, wax beans, edamame, cherry tomatoes, basil and salt; toss gently to coat.
Divide the salad between 8 plates. Top each with a burrata quarter and serve.