Pound the chicken breasts lightly with a meat mallet to flatten them a bit and make it easier to thread onto skewers. Cut the chicken breasts into strips lengthwise, about 1 ½ inches wide.
Whisk together the cup of Greek yogurt, half cup of olive oil, mustard, cayenne, cumin, savory, petimezi, orange juice, salt and pepper in a large bowl. Swirl the chicken strips in this mixture, turning to coat. Refrigerate, covered for at least an hour or up to 8.
Soak 12 – 16 wooden skewers in water for a half hour before using them. Alternatively, use metal skewers.
Preheat the grill or broiler to medium. Thread the marinated chicken strips onto the skewers. Grill or broil the skewers, turning to cook on all sides. They should take about 10 – 12 minutes in total.
While the chicken is cooking, make the avocado tzatziki: Mix the yogurt, cucumber, avocado, garlic, lemon juice, salt, hot sauce and olive oil together. Taste and adjust seasoning as desired.
Remove the skewers and serve on a platter with a bowl or two of the avocado tzatziki as accompaniment.