Place oil in a large skillet. Place skillet over medium heat.
When oil is hot and shimmering, add chopped red peppers and chopped onions to skillet. Sauté until just tender. Set aside.
Prepare the charcoals for grilling with the coals glowing hot.
Grill chicken to an internal temperature of 165°F.
Remove from grill and set aside to cool. When cooled to touch, shred with a fork or pull apart with hands or, alternately, cut into small cubes.
Roll each piece of dough out to about 1/8 inch thick.
Brush lightly with olive oil on both sides.
Place doughs directly on the grill grate.
Checking with a spatula, grill until the bottom is golden brown and the top starts to bubble, about 5 minutes.
Using 2 spatulas, turn dough over.
Quickly spread 2 ounces Mozzarella cheese (1/4 of total) over each dough.
Scatter chicken, peppers, onions and chives over each dough.
Place lid on grill. Grill for approximately 6 minutes, or until the bottom is golden brown and cheese is melted.
Remove from grill, cut into wedges and serve hot.