Trim the ends of the zucchini, then use a vegetable peeler to peel into thin, flat noodles. Repeat with the sweet potato and daikon radish. Set aside.
In a wok or large pan over medium-high heat, combine the brown sugar, fish sauce, tamarind paste, 3 cloves of garlic, and the water. Cook until the sugar has dissolved and the sauce is slightly thickened, about 3 minutes. Remove the pan from the heat and transfer the pad thai sauce to a bowl until ready to use. Wipe out the pan.
Heat a drizzle of coconut oil in the same pan over medium-high heat. Add the chicken thighs and cook for 4 minutes on each side, until browned and cooked through. Remove the chicken from the pan and set aside.
Add a bit more coconut oil to the pan, if needed, then add the shallot, chili paste, and remaining 3 cloves of garlic. Cook until the shallots are softened and the aromatics are fragrant, 2 minutes.
Add the peanuts and bean sprouts, and cook for 1 minute, until the sprouts are softened. Add half of the reserved pad thai sauce and cook until thickened, 3 minutes.
Add the vegetable noodles and chicken, and stir to coat with the sauce, adding more as needed. Add the cilantro, toss, and cook for 30-60 seconds, until the noodles are just warmed through but not soggy.
Serve immediately garnished with more cilantro, bean sprouts, peanuts, and lime wedges.