In a small bowl, whisk together half of the olive oil, half of the lemon juice, dried oregano, garlic powder, sea salt, and black pepper to create a marinade.
Place the chicken breast in a shallow dish and coat thoroughly with the marinade; let it sit for at least 10 minutes.
Preheat a grill or grill pan over medium-high heat and cook the chicken for 6 to 7 minutes per side, or until the internal temperature reaches 165°F.
While the chicken is grilling, chop the romaine lettuce, slice the cucumber into half-moons, dice the red bell pepper, and thinly slice the radishes.
In a large mixing bowl, combine the romaine, cucumber, bell pepper, radishes, and shredded carrots.
Whisk the remaining olive oil and lemon juice together and toss with the vegetable mixture until lightly coated.
Remove the chicken from the grill and let it rest for 5 minutes before slicing into thin strips.
Top the salad with the grilled chicken strips and garnish with fresh avocado slices.