If your chicken breasts are very thick, slice them in half horizontally to create thinner cutlets. In a large bowl or resealable bag, combine the olive oil, white wine vinegar, minced garlic, Italian seasoning, smoked paprika, salt, pepper, and red pepper flakes.
Add the chicken to the marinade and turn to coat completely. Let it marinate for at least 30 minutes at room temperature, or cover and refrigerate for up to 4 hours for more flavor.
While the chicken marinates, prepare the pesto mayo. In a small bowl, stir together the mayonnaise, basil pesto, and lemon juice until smooth. Season with a small pinch of salt and pepper if needed. Set aside.
Preheat your grill to medium-high heat, around 400°F (200°C). Clean and lightly oil the grates.
Place the chicken on the grill. Cook for 5-7 minutes per side, flipping once. The chicken is done when it's opaque all the way through and a thermometer inserted into the thickest part reads 165°F (74°C).
During the last minute of cooking, top each piece of chicken with a slice of cheese and let it melt.
Remove the chicken from the grill and let it rest on a cutting board for 3-5 minutes. This helps keep it juicy.
While the chicken rests, slice the ciabatta rolls and toast them on the grill for about a minute until lightly golden.
To assemble, spread a generous layer of pesto mayo on the toasted rolls. Add a piece of cheesy chicken, a few slices of tomato, and a handful of arugula. Serve immediately.