Preheat a grill to medium. Prepare the chicken: With your knife parallel to the cutting board, slice each chicken breast in half horizontally without cutting all the way through; open like a book. Cover with plastic wrap and pound until 1/2 inch thick.
Make the filling: Cook the rice vermicelli as the label directs; drain. Toss with 1 tablespoon vegetable oil, the rice vinegar, lime zest, Sriracha and a pinch of salt in a bowl. Whisk the lime juice with the remaining 2 tablespoons vegetable oil in a separate bowl; season with salt and pepper.
Season the pounded chicken on both sides with salt and pepper and brush with the lime oil; top with the noodles, carrot, bell pepper and scallion, leaving a 1-inch border around the edges. Roll up the chicken toward the pointy end; tie with 3 pieces of kitchen twine.
Brush the grill grates with vegetable oil. Grill the chicken breasts, turning, until cooked through, 18 to 20 minutes, brushing with the sweet Thai chili sauce during the last 2 minutes of cooking. Transfer to a cutting board and let rest 5 minutes, then remove the twine and slice.