Cook the bacon in a large skillet over medium heat, turning a few times until crisp, about 8 minutes. Transfer to a paper towel-lined plate to drain, then cut each slice in half. While the bacon is cooking, pulse the basil, olive oil, lemon juice, 1 tablespoon water, 1/4 teaspoon salt, and pepper to taste in a mini food processor (or finely chop the basil and whisk with the other ingredients) until smooth. Add the mayonnaise and pulse until just combined. Toast the bread and spread evenly with the basil mayonnaise. Divide the peaches, bacon and arugula among half of the bread slices; top with the remaining bread slices. Serve with cornichons.