Heat vegetable oil in a large wok or skillet over high heat. Add the shallots and stir until just starting to turn golden, about 30 seconds. Add the garlic and chiles and continue stirring for another 15-30 seconds, being careful not to let it burn. Quickly add the ground chicken and use a wooden spatula to break it up into little bits. Season with salt and plenty of black pepper, then use the back of the spatula to press down on the chicken to help it get brown and crisp, about 10 minutes.
In a small bowl, whisk together the brown sugar, chicken broth, soy sauce, fish sauce and lime juice until the sugar is dissolved. Pour it into the pan with the chicken, and stir. Allow it to cook until the liquid has mostly evaporated, about 5 minutes. Stir in half of the scallions and remove from the heat.
To serve, transfer the chicken to a serving bowl and top with the remaining scallions. Serve family style at the table along with the pickled carrots and shallots, puffed rice noodles, lettuce leaves, fresh herbs and Sriracha.