Melt unsalted butter in a skillet over medium-high heat. Add corn, then season with kosher salt. Cook until the corn is golden. Transfer to a blender.
To the blender, add chopped tomatoes, chopped peaches, fresh basil leaves, extra virgin olive oil, kosher salt, and white pepper. Blend until very smooth, then taste and adjust seasoning as needed, adding more olive oil or salt to taste.
Just before serving, melt the remaining butter in a skillet. Add crumbled cornbread, then season lightly with kosher salt. Cook.
Divide the soup, then top with crispy cornbread, a drizzle of extra virgin olive oil, thinly sliced chives, a few basil leaves, and flaky sea salt. Enjoy!