Place onions into the bottom of a multi-functional pressure cooker (such as Instant Pot). Pour melted butter and broth on top.
Mix garlic, poultry seasoning, salt, pepper, paprika, and smoked paprika together in a small bowl. Stir into onions. Place chicken on top of the onions.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 9 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 40 minutes.
Transfer chicken and onions to a plate. Select Sauté function to bring juices to a simmer, about 5 minutes.
Sift flour into the pot and stir until no lumps remain and sauce has thickened a bit, about 5 minutes. Add sour cream by the spoonful, stirring to incorporate. Cancel Sauté function. Return chicken and onions to the pot. Select Warm function to heat through, about 5 minutes. Serve.