Preheat your barbecue to medium-high.
Combine first 6 ingredients in a bowl. Shape the turkey into 12 small, 1/2" thick patties. Grill the patties 3 minutes per side, or until cooked through. Turn the heat to low, top each patty with a piece of cheese, close the lid and cook 1 minute more. Spread the buns with the pesto flavoured mayonnaise. Set a turkey patty on each bottom bun. Top with cherry tomato slices, add top buns, and serve.