Pour grapeseed oil into an electric pressure cooker (such as Instant Pot). Add carrots, leeks, shiitake mushrooms, and onion; cook and stir on the Sauté setting until softened, about 5 minutes.

Season chicken thighs with salt and pepper; add to the pot. Pour in chicken broth. Seal and cook on the Soup setting for 7 minutes. Release pressure naturally for 10 minutes according to the manufacturer's instructions. Cover the vent with a dishtowel and release remaining pressure with the quick-release method.

Remove chicken thighs and shred with 2 forks on a cutting board; return to the pot and add miso paste, garlic, soy sauce, ginger, and Sriracha. Cook and stir on the Sauté setting until miso paste dissolves, about 5 minutes. Stir in cabbage and bok choy; cook until softened, about 5 minutes.