Rinse the farro under cold water. Place it in a saucepan with the water and half the salt. Bring to a boil, then reduce heat and simmer for 25 minutes until tender. Drain any excess liquid and set aside.
In a bowl, mix the olive oil, juice of one lemon, minced garlic, oregano, remaining salt, and pepper. Add the chicken breasts and coat them well. Let the chicken marinate while you prepare the tzatziki.
In another bowl, stir together the Greek yogurt, grated cucumber, dill, and juice of the remaining lemon. Add a pinch of salt if needed and set the tzatziki aside.
Heat a grill or grill pan over medium high heat. Grill the chicken for 6 to 7 minutes per side until it reaches an internal temperature of 165 degrees. Remove the chicken and let it rest for 5 minutes before slicing.
Spoon the cooked farro onto plates. Top with sliced chicken and a generous dollop of tzatziki.