Let roast come to room temperature, about 1 hour. Season with salt and pepper.

Heat oil in a multi-functional pressure cooker (such as Instant Pot®) on Saute mode. Add roast; sear on all sides until well browned, about 3 minutes per side. Remove from pot.

Add more oil if necessary; add onion and cook until softened, about 5 minutes. Add garlic and saute until fragrant, about 30 seconds. Pour wine into the pot and bring to a boil while scraping the browned bits of food off the bottom with a wooden spoon.

Add roast, beef broth, soy sauce, vinegar, salt, parsley, thyme, and pepper to the pot; stir. Close and lock the lid. Select Meat/Stew function according to manufacturer's instructions; set timer for 60 minutes.

Release pressure using the natural-release method according to manufacturer's instructions, at least 20 minutes.

Remove roast and transfer to a bowl. Strain broth through a fine-mesh strainer. Skim the fat from the top and add browning sauce. Shred beef using 2 forks and add 1 cup of the broth. Divide remaining broth between individual ramekins.

Place provolone cheese on 1/2 of the ciabatta slices and toast all the bread until cheese has melted. Add beef to each sandwich and dip each into broth to serve.