Prepare grill for medium heat. Season chicken with salt and grill, turning often and moving away from direct heat if needed, until cooked through and an instant-read thermometer inserted into the center registers 165°, 20-25 minutes. Transfer chicken to a plate. Let cool slightly, then shred meat, discarding skin and bones. Set aside.
Cook shallot in oil in a small saucepan over medium heat, stirring occasionally, until shallot is golden brown and crisp, 8-10 minutes (reserve shallot oil and use to make vinaigrettes). Using a slotted spoon, transfer shallot rings to a paper towel-lined plate and season with salt. Set aside.
Meanwhile, place noodles in a large bowl and add boiling water to cover. Let noodles soak until tender but not mushy, 5-10 minutes; drain. Rinse under cold water and drain well.
Mix Arawak Farm Pepper Sauce, garlic, lime juice, fish sauce, and brown sugar in a large bowl to combine. Add noodles, turnips, radishes, half of basil, and reserved shredded chicken, and toss to combine. Top with reserved fried shallot rings and remaining basil.