Combine soup and stew seasoning in a small bowl until a creamy paste forms.

Turn on a multi-functional pressure cooker (such as Instant Pot) and select Sauté function. Add chuck roast; cook until juices start to render and outside is browned, about 5 minutes per side.

Add soup paste mixture, broth, onion, and mushrooms. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 40 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.

Add potatoes and carrots. Close and lock the lid. Return cooker to high pressure according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure using the natural-release method according to manufacturer's instructions, 10 to 30 minutes.

Transfer roast to a large bowl and shred with 2 forks; remove any excess fat.

Strain remaining liquid through a sieve into a separate bowl. Whisk liquid until well combined and emulsified, about 1 minute.

Serve shredded roast on top of vegetables; pour gravy over top.