Whisk the water with the lemon juice and salt until the salt dissolves. Place the chicken breasts into a resealable gallon sized bag, then pour the brine over the chicken. Add the peppercorns, garlic, and bay leaf to the bag. Lightly twist the oregano spring between your palms to release the oils, then add to the bag. Press out the air, and place in the fridge to brine for at least 2 hours and up to 8.
When you're ready to eat, heat a grill or grill pan over medium high heat.
Remove the chicken from the brine and pat dry. Brush both sides of the chicken with the olive oil, then place on the grill. Grill the chicken 6-7 minutes, then flip. Grill on the other side an additional 6-7 minutes until the internal temperature reaches 165* with an instant read thermometer. Remove from heat and set aside to rest for at least 5 minutes.
Place the mixed greens onto a platter or into a large salad bowl. Top the greens with the artichoke hearts, cherry tomatoes, kalamata olives, persian cucumbers, and fresh mozzarella.
Place all of the vinaigrette ingredients into a blender. Blend until the oregano is completely pulverized the the vinaigrette is smooth.
Slice the chicken against the grain into 1/2" slices, and place over the salad base. Drizzle with oregano vinaigrette and sprinkle with a tablespoon of fresh oregano if desired. Serve immediately. This salad keeps in the fridge, undressed, for up to three days.