Preheat the oven to 350°F and line a baking sheet with parchment paper.
In the bottom of a large bowl, make the dressing by mixing together 3 tablespoons of olive oil, the vinegar, mustard, salt, shallot, and garlic. Set aside.
Arrange the bread cubes on the baking sheet, drizzle with olive oil, and toss to coat. Bake until toasted, about 8 minutes.
To the large bowl, add the (raw) corn, tomatoes, toasted bread, arugula, a few pinches of red pepper flakes, and toss. Season to taste with flaky sea salt (or regular sea salt). Spread onto a platter and top with the basil and shaved cheese, if using. Serve.
Notes
Make-ahead note: If you’re not serving this right away, mix every ingredient together except the bread, arugula, basil, and cheese. Add the bread and arugula 15 minutes before serving so that the bread does not get too soggy, and the arugula does not wilt. Top with the basil and cheese as you’re ready to serve.