In a small bowl, whisk together half of the olive oil, half of the lemon juice, the dried oregano, 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 15 minutes to absorb the flavors.
Preheat a grill or cast-iron grill pan over medium-high heat and lightly grease with a tiny bit of oil if needed.
Grill the chicken for 6 to 7 minutes per side, or until the internal temperature reaches 165°F and the outside is nicely charred.
While the chicken rests for 5 minutes, prepare the salad base by combining the mixed greens, sliced cucumber, and halved cherry tomatoes in a large bowl.
Whisk the remaining olive oil and lemon juice together and toss with the salad greens.
Slice the rested chicken into strips and arrange them over the salad.
Top with the sliced avocado and an extra pinch of black pepper if desired before serving.