Combine lime juice, soy sauce, onion, cilantro, peanut butter, oil, brown sugar, ginger, lime zest, red pepper flakes and garlic in a stockpot. Stir in shredded turkey. Heat over low heat, stirring occasionally.
Unroll pizza dough and pat into a rectangle, approximately 10 x 13 inches. Cut dough into 4 equal pieces.
Reduce grill heat to low. Brush grill rack with olive oil. Using the indirect grilling method, slide pizza crusts on grill and cook until golden, about 3 to 4 minutes, until dough is puffy and lightly browned.
Turn pizza crusts over. Top browned side of piza crusts with heated turkey mixture. Sprinkle with green onions, carrot sticks and cilantro. Sprinkle with cheese, if using.
Cover with grill lid or tent with foil. Heat pizzas about 5 minutes or until crusts are cooked on bottom, cheese melts and pizzas are hot.