Preheat the oven to 350ºF.
Spread the bread cubes out on a baking sheet. Toss with 1 tablespoon of the olive oil and 1/2 teaspoon kosher salt (add a bit more oil as needed to coat the bread). Bake in the preheated oven for 15 minutes, until toasted and crisp.
Meanwhile, transfer the chopped tomatoes and sliced peaches to a colander and fit the colander over a bowl. Sprinkle the fruit with 1/4 teaspoon kosher salt and gently toss to combine. Allow the fruit to rest while the bread is toasting, at least 15 minutes.
Remove the colander and transfer the peaches and tomatoes to a large serving bowl. Keep the peach and tomato juices that have accumulated in the bowl. To the juices, add the 2 tablespoons of vinegar. Slowly drizzle in the remaining 4 tablespoons of olive oil, whisking as you go to emulsify. Season the vinaigrette with salt and pepper to taste.Or, you can transfer the peach/tomato juices to a lidded jar and add the olive oil/vinegar, and shake vigorously until emulsified.
To the peaches and tomatoes in the serving bowl, add the sliced onions and basil, and gently toss to combine. Add the toasted bread cubes and vinaigrette, and gently toss again to coat.
Rest for 30 minutes (or up to 2-3 hours) before serving, tossing the salad occasionally. Taste, and season with a bit of salt and pepper, as needed.