Bring a pot of water to a boil. Meanwhile, heat the vegetable oil in a large skillet over medium-high heat. Season the chicken on both sides with salt and pepper. Add skin-side down to the skillet and cook until dark brown and crisp, about 8 minutes; flip and brown the other side, 2 minutes. Remove to a plate and wipe out the skillet.
Off the heat, add 2 tablespoons brown sugar to the skillet and sprinkle the fish sauce over the sugar. Return the skillet to medium heat, stir and bring to a simmer. Add the coconut milk and 1 teaspoon vinegar; stir to combine. Return the chicken skin-side up to the skillet and gently simmer in the sauce, turning occasionally, until cooked through, about 25 minutes. Add a splash of water occasionally if the sauce gets too thick.
Meanwhile, halve the cucumber lengthwise and scoop out the seeds. Cut the cucumber and carrot into thin matchsticks. Toss with the remaining 2 tablespoons vinegar, 1 teaspoon brown sugar and 3/4 teaspoon salt in a medium bowl.
Remove the boiling water from the heat, add the rice noodles and stir to loosen. Let sit until tender, 2 to 4 minutes, then drain and rinse under cold water. Shake out as much water as possible.
Divide the noodles and chicken among plates. Spoon the coconut sauce on top. Serve with the vegetables; top with the mint.