Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Adjust to Low if necessary. Pour in sesame oil and olive oil. Add ginger and garlic to the hot oil; cook until fragrant without browning, about 1 minute. Add chicken broth, wontons, salt, and pepper. Break apart any wontons that have frozen together. Close and lock the lid.

Select Soup function; set timer for 2 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Mix in bok choy and soy sauce and let sit, stirring occasionally, for a few minutes. Taste and season with additional salt if necessary.

Ladle soup into bowls and top with green onions.