Dice the red onion and place into a bowl. Add the pesto, feta cheese, chopped basil, salt, pepper and parsley flakes. Add the ground chicken and mix until combined. Heat a panini grill, griddle or outdoor grill to medium-high heat. Brush the griddle or grill with a little olive oil to prevent sticking. Form the chicken mixture into 6 patties. Place the patties on the grill to cook. While they are cooking, slice the tomatoes, the other part of the red onion, and the lettuce leaves for assembly of the burgers. In a small bowl, combine the mayonnaise and the pesto (add more pesto if you like a lot of pesto). Spread this sauce generously on both sides of the bun. When one side of the burger is nicely browned, flip to the other side and continue cooking until the center is cooked through. Place the chicken burgers on each of the hamburger buns. Add the sliced tomatoes, red onion and the lettuce on the burgers and serve.