For the sauce: In a small bowl, stir together the fish sauce, vinegar, agave, lime juice, chile, salt and 1/3 cup water. Set aside to let the flavors come together.
For the spring rolls: Grate the carrot, ginger and shallot on the large holes of a box grater, and transfer to a large bowl. Add the chicken, garlic, shiitakes, cilantro, mint, chiles, egg, fish sauce, agave and salt. Mix together until well combined, using a rubber spatula. Cover with plastic wrap and let sit at room temperature for 20 minutes.
Preheat the broiler to high. Lightly spray a baking sheet with nonstick spray.
With dampened hands, shape the chicken mixture into oblong patties, about 2 tablespoons each. Place the patties on the prepared baking sheet and broil until cooked through and beginning to brown, 8 to 10 minutes. Remove from the oven and let cool slightly.
To assemble: Arrange the patties on a platter with the lettuce, basil, cilantro and mint leaves, so everyone can make there own spring roll. Serve the sauce on the side for dipping.
Recipe courtesy of Giada De Laurentiis