Slice the bagel in half, length-wise (like a hamburger bun), then thinly slice each half into thin half-moon slices (approximately ¼-inch thick).
Spread the bagel slices in a single layer on a baking sheet and and toast in the oven for 10 minutes, tossing halfway through. Remove these bagel "croutons" from the oven and allow them to cool. (They'll continue to harden and get crunchy as they cool.)
Combine the dressing ingredients in a mason jar, seal the lid, and shake for 30 seconds or until combined (alternatively, you can whisk ingredients together in a small bowl).
To a large bowl, add the blueberry bagel croutons, fresh blueberries, grape tomatoes, peach slices, and the dressing. Toss to combine. Cover bowl and refrigerate for at least 30 minutes to allow the flavors to soak together.
Prior to serving, julienne the fresh basil and add the freshly chopped basil and the arugula to the salad. Toss to combine.
Serve immediately, or refrigerate further until serving.