Set Instant Pot to sauté. When hot, add butter and allow to sizzle. Add eggs cook for 1-2 minutes until eggs begin to firm. Flip eggs and cook an additional 30 seconds. Remove, cut into thin strips and set aside. Cancel saute, remove inner liner, wipe it clean, then return it to the Instant Pot.
Set Instant Pot to sauté. When hot, add olive oil and garlic. Cook 1-2 minutes. Add beef and brown on each side.
In a small bowl, add vinegar, hoisin and soy sauces, sesame oil, ginger, and brown sugar. Whisk together and reserve 1/4 cup. Pour remainder of sauce into Instant Pot, covering the beef. Set Instant Pot to high pressure for 35 minutes. When the cook time ends, let the pressure naturally for 15 minutes.
After 15 minutes, turn valve to venting to release remaining pressure. When pressure is fully released, remove the lid. Using two forks, shred beef.
Add mushrooms and 2 cups of the coleslaw mix to the beef in the Instant Pot. Secure lid, turn valve to sealing, press Manua and set Instant Pot to high pressure for 3 minutes. Allow to release pressure naturally for 5 minutes, then quick release remaining pressure. Turn off Instant Pot.
Serve over rice or add 2 tablespoons of Moo Shu Beef to individual lettuce cups. Top Moo Shu Beef with a few slices of egg and coleslaw mix. Drizzle sauce. Enjoy!