Place the onion and ginger on a baking sheet. Broil, turning halfway through, until nicely charred but not burned. I broiled mine for 10 minutes.
Set the Instant Pot to Sauté. Add the pho spice mix and toast for 30–60 seconds until fragrant. Transfer the spices to a spice bag or cheesecloth.
Add the rotisserie chicken carcass and bones to the Instant Pot.
Pour in the chicken broth. Add the charred onion, ginger, fish sauce, chicken pho powder, sugar, and the spice bag.
Seal the lid and cook on High Pressure for 30 minutes. Allow the pressure to release naturally for 10–15 minutes, then carefully release any remaining pressure.
Remove and discard the chicken bones, onion, ginger, and spice bag. Strain the broth through a fine-mesh sieve and season with additional salt or MSG, if needed.
Cook the pho noodles according to the package directions.
Divide the noodles among serving bowls. Top with shredded rotisserie chicken, then ladle the hot broth over the top. Garnish with green onions, lime wedges, fresh herbs, hoisin sauce, and sriracha.