Season the chicken breast evenly on both sides with sea salt, black pepper, and dried oregano.
Heat a grill or cast-iron skillet over medium-high heat and add one tablespoon of the olive oil to coat the surface.
Place the chicken on the grill and cook for 6 to 8 minutes per side until golden brown and the internal temperature reaches 165°F.
While the chicken cooks, whisk together the remaining half tablespoon of olive oil, balsamic vinegar, and Dijon mustard in a small bowl to create the vinaigrette.
Slice the cucumber into thin rounds and halve the cherry tomatoes.
In a large salad bowl, combine the mixed greens, cucumber, and cherry tomatoes, then toss thoroughly with the prepared vinaigrette.
Remove the chicken from the heat and let it rest for 3 minutes to lock in the juices.
Slice the chicken into strips and serve immediately over the fresh garden salad.