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.