Halve the cherry tomatoes and shave or grate the Parmesan. Rub the chicken fillet on both sides with salt, pepper and Italian herbs. Whisk the olive oil and lemon juice together in a small bowl to make a smooth dressing and set aside.
Heat a grill pan over medium-high heat and wait until it is very hot. Place the seasoned chicken fillet in the pan and grill the meat for six to eight minutes per side until fully cooked. Remove the chicken from the pan and let the meat rest for five minutes on a cutting board so the juices are retained.
Slice the rested chicken fillet into thin strips. Put the arugula together with the halved cherry tomatoes in a large mixing bowl. Pour the prepared dressing over the vegetables and toss thoroughly so the leaves are well coated.
Spread the dressed salad in an even layer on a flat plate. Arrange the warm grilled chicken strips on top of the arugula and tomatoes. Finally sprinkle the Parmesan over the dish and serve immediately.