Pat the chicken fillets dry with paper towels. Rub the meat with 15 milliliters of extra virgin olive oil, salt, and freshly ground black pepper. Rinse the cucumber, cut it in half lengthwise, and remove the seeds. Slice the cucumber into thin half-moons. Make a dressing by whisking the remaining 30 milliliters of olive oil with the freshly squeezed lemon juice, Dijon mustard, and finely chopped garlic. Season to taste with salt and pepper.
Heat a grill pan over medium-high heat. Place the chicken fillets in the hot pan and grill them 6 to 8 minutes per side. Check that the chicken is cooked through and no longer pink inside. Remove the chicken from the pan and place it on a cutting board.
Let the meat rest for 5 minutes. In the meantime, place the mixed greens and sliced cucumber in a large bowl. Slice the rested chicken fillets into strips about one centimeter thick and add them to the vegetables.
Pour the prepared dressing over the salad and chicken. Toss everything together gently so all ingredients are covered. Serve immediately.