Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Add butter and melt. Stir in garlic and saute until fragrant, about 1 minute. Stir in orzo and 1 1/2 cups chicken stock; mix well. Cancel Saute function. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 6 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 mushrooms and switch to Saute mode.
Stir in Parmesan cheese, garlic salt, and rosemary. Stir in remaining 1/2 cup chicken broth 1 tablespoon at a time until the orzo takes on a creamy texture, about 3 minutes.
