1. In a medium bowl, combine tomatoes, parsley, thyme, 1/4 teaspoon salt, 1/8 teaspoon pepper, 2 tablespoons olive oil, 1/4 cup onion, and vinegar. Set aside.Make Burgers::
2. In a separate bowl, combine mayonnaise, mustard, and cayenne pepper. Set aside.
3. Heat 1 tablespoon oil in a large nonstick skillet over medium heat. Add mushrooms and remaining onions and cook, stirring occasionally, about 7 minutes, until onions are translucent. Stir in garlic and cook 1 minute more.
4. Place half of chicken in a food processor with a metal blade. Pulse a few times until chicken is ground. Add egg, breadcrumbs, remaining salt, and pepper. Pulse twice to blend. Add remaining chicken and pulse just until chicken is cut into small cubes but not ground. Turn mixture out into a bowl and stir in mushroom-onion mixture. With clean, wet hands, shape into 4 patties. Set aside.
5. In same skillet, heat remaining oil over medium heat and cook chicken burgers about 5 minutes per side, until cooked through. Spread mayonnaise mixture on rolls and serve burgers with tomato relish.