Season the chicken breast evenly with half of the sea salt, black pepper, and dried oregano.
Grill the chicken over medium-high heat for approximately 6 to 7 minutes per side until the internal temperature reaches 165 degrees Fahrenheit.
While the chicken cooks, prepare the vinaigrette by whisking together the extra virgin olive oil, lemon juice, Dijon mustard, and the remaining salt, pepper, and oregano in a small bowl.
Thinly slice the cucumber into half-moons, halve the cherry tomatoes, and finely slice the red onion.
Place the mixed baby greens in a large serving bowl and toss gently with half of the prepared vinaigrette.
Allow the grilled chicken to rest for 3 minutes before slicing it into thin strips.
Arrange the sliced chicken, cucumber, tomatoes, and red onion over the greens and drizzle with the remaining vinaigrette before serving.