Heat the olive oil in a medium skillet over medium heat. Season the diced chicken breast with salt and black pepper. Cook the chicken until nicely browned and the internal temperature reaches 165°F (74°C). Remove from heat and let it rest for a few minutes.
Spread the herb mayo onto each tortilla. Top with cooked chicken, cucumber, tomato, capers, feta cheese, and lettuce.
Fold in the sides of each wrap, then roll it up tightly from the bottom to form a secure wrap. Cut in half.
Stir together lemon juice, water, and sugar to taste. Serve over ice if desired.