Heat grill to medium. Rub chicken with 1/2 tsp olive oil and sprinkle with 1/4 tsp each salt and pepper. Grill 8–10 minutes per side or until internal temperature reaches 165°F. Transfer to cutting board and let rest for 10 minutes before slicing.
While chicken is grilling, place lemon halves and cut-side baguettes on grill until charred, about 2–3 minutes. Transfer to cutting board and rub cut side of baguette with garlic clove.
Squeeze 2 tablespoons juice from grilled lemon into a bowl. Whisk in mayonnaise, Dijon mustard, and anchovy paste. Gradually whisk in remaining 2 tablespoons olive oil until emulsified.
In a large bowl, toss romaine and celery with 1/4 cup of the dressing and 1/4 tsp black pepper. Add Parmesan and another 1/4 tsp black pepper.
Drizzle remaining dressing on both halves of the baguette. Layer with sliced grilled chicken, Caesar salad, and tomato slices.
Cut each baguette half in half again to yield 4 sandwiches. Serve immediately or wrap for later.