For the sauce: In a bowl, whisk together the peanut butter, water, soy sauce, garlic, and red pepper flakes and set aside.
For the rolls: Fill a shallow baking dish or pie plate with hot water. Soak a rice paper wrapper until softened, about 30 seconds. Place it onto your work surface. Along the bottom edge, layer one-eighth of the noodles, cucumbers, carrots, and mint leaves. Fold the bottom edge up over the filling. Fold in the sides and continue rolling up from the bottom. Repeat with remaining rice paper wrappers.
Serve with the dipping sauce.