Preheat your oven to 425°F. Line two rimmed baking sheets with aluminum foil. Place a wire rack on one of the baking sheets for the pork belly.
Combine the potatoes, zucchini, and Fresno chiles in a large bowl. Add 1 tablespoon of the oil and a pinch of salt and toss to coat, then spread in a single layer on the lined baking sheet. Rub the pork belly with the remaining 1 tablespoon of oil and season on both sides with salt. Place it on the wire rack. Roast the vegetables and pork belly in the oven until the pork belly is cooked through and tender, about 45 minutes, tossing the vegetables at the halfway mark.
Meanwhile, combine the chipotles and their sauce, the lime juice, 5 teaspoons of sugar, and the soy sauce in a blender and blend until smooth. Transfer to a small bowl and set aside. No need to wash the blender.
When the skin of the Fresno chiles has blackened and charred, about 25 minutes, remove them from the oven (return the remaining vegetables to the oven to continue roasting with the pork belly) and transfer to a ziplock bag. Seal the bag and allow the chiles to steam until their skins have loosened, about 10 minutes. Remove and discard the skins and stems (you may want to wear gloves to protect your hands), and transfer the chile flesh to the blender, along with the garlic, the remaining 1 teaspoon of sugar, 1 teaspoon of salt, and ½ cup of water. Blend until smooth, then transfer to a small bowl, cover, and set aside.
When the pork belly and vegetables are done, remove them from the oven and let rest. Bring a large pot of water to a boil over high heat. Add the noodles and cook according to the package directions; drain well. Bring the beef stock to a simmer in a medium pot over low heat and hold. Slice the pork belly into ½-inch-thick slices.
Place 1 tablespoon of the chipotle mixture and 1 teaspoon of the Fresno mixture into each of four warmed bowls, then ladle 2 cups of beef broth into each. Divide the noodles among the bowls and top each with a few slices of the pork belly and some of the vegetables. Serve piping hot, with the arugula in a bowl and the remaining chipotle and Fresno mixtures alongside as condiments.