In a bowl, combine salt, pepper, olive oil, lemon zest and juice, and basil. Whisk to combine. Add zucchini ribbons and red bell pepper and gently toss to coat with dressing.
Spread one quarter of Boursin cheese onto one half of each sandwich. Divide shredded chicken among the four sandwiches, placing on top of cheese. Top chicken with red bell pepper and zucchini ribbons, drizzling any remaining dressing over the vegetables. Cut each sandwich on the diagonal and serve.