Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add olive oil, followed by onion, carrot, and garlic; cook until onion is translucent, 5 to 7 minutes. Add ground chicken and ground sausage and cook until browned, 5 to 10 minutes.
Stir in cabbage, water, tamari, honey, ginger, pepper, and salt. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, 10 to 15 minutes. Stir and add sesame oil.