Add pork to the insert of the Instant Pot. Pour in chicken broth, 1/2 cup soy sauce, rice vinegar, sake, and fish sauce. Stir in sambal oelek, lime juice, Chinese five spice, black pepper, sesame oil, mushrooms, 1 Tbsp brown sugar, and dark soy sauce. Seal the Instant Pot. Set to Manual Pressure and cook on High for 90 minutes. Quick release when finished.
Remove pork from the Instant Pot insert. Cover the Instant Pot and set to Warm.
Place pork on a cutting board and lightly shred using two forks.
Place a large skillet over medium-high heat. Add sesame oil. Once the oil is hot, add pork and spread it out in an even layer. Sprinkle the pork with brown sugar and 3 Tbsp soy sauce. Cook without stirring for 2 minutes. Stir and continue to cook the pork until it starts to crisp, about 5 minutes total. Remove from heat.
Add the ramen noodles to the Instant Pot and allow to cook through, about 5 minutes. Stir in half of the pork.
Serve the soup immediately in bowls. Top with remaining pork, eggs, carrots, jalapeƱo slices, cilantro, green onion, and sesame seeds.