For the Pickled Carrots: Place the vinegar, sugar, and salt in a small, nonreactive saucepan over medium heat. Cook, stirring occasionally, until the sugar and salt have dissolved. Remove from the heat, add the carrots, and stir to coat them in the pickling mixture. Let stand at room temperature until the carrots have softened, at least 30 minutes or overnight in the refrigerator. Drain well and refrigerate in a covered container until ready to use.
For the Chile Mayonnaise:. Combine the mayonnaise and Sriracha in a small bowl. Cover and refrigerate until ready to use.
For the Burgers: Place all of the ingredients except the chicken and oil in a large bowl and stir to combine. Add the ground chicken and mix with your hands until just evenly combined.
Divide the meat mixture into 8 equal portions. Shape each portion into an even 1/2-inch-thick patty, wetting your hands with water as needed if the mixture is sticky. Transfer the patties to a baking sheet and refrigerate while you prepare the grill.
Heat a grill pan or outdoor grill to medium high (about 375°F to 425°F). When the grill is ready, use tongs to rub the grate with several layers of paper towels dipped in vegetable oil.
Using a flat spatula, transfer the reserved patties to the grill, close the grill, and cook undisturbed (do not press down on the patties) until grill marks appear on the bottom, about 3 to 4 minutes. Using a clean spatula, flip the patties, close the grill, and cook until the patties are cooked through and the juices run clear, about 3 to 4 minutes more.
Remove to a clean plate and let rest in a warm place or tent loosely with foil for 5 minutes.
To assemble: Spread about 1 tablespoon of the chile mayonnaise on each top and bottom roll (2 tablespoons per sandwich). Place 2 of the chicken patties side by side on each bottom roll. Divide the cilantro leaves, sliced jalapenos, pickled carrots, and romaine lettuce evenly among the sandwiches and cover with the top rolls. Serve immediately.