Set the Instant Pot (IP) to saute mode on the more setting. Add the butter and olive oil and heat until the butter is completely melted. Add in the onions and sugar and stir well to combine.
Cook the onions on the more setting for 10 minutes then change the saute setting on the IP to "less." Contine cooking uncovered for 40 minutes, until the onions are tender and caramelized. Do not over stir - I stirred once every 5-10 minutes to prevent sticking.
Use tongs to remove the caramelized onions from the IP and set aside. Use the white wine to deglaze the bottom of the IP. Then add the chicken breasts to the IP along with salt and pepper, to taste.
Add in the whole garlic, parsley and French onion soup. Stir in the beef broth.
Place the lid on the IP and make sure to set the vent to "sealing." Then put the IP in Pressure Cook mode on the more setting. The IP will start to pressurize over the next 5-10 minutes. When pressurized, the IP will begin the timer.
Meanwhile, preheat the oven to 350 F | 175 C. Slice and butter the hoagie rolls and place slices of provolone on top. Toast the rolls for 5-10 minutes, until the cheese is starting to sweat and melt. Do not over melt the cheese - it should act like a barrier between the meat and bread.
When the IP has finished cooking, do a quick release or let the IP depressurize naturally. Set the IP to saute mode for 15 minutes and simmer until the liquid has reduced.
Use a ladle to spoon out dipping portions of the au jus liquid in ramekins or small bowls.
Spread the shredded chicken over the rolls and top with caramelized onions, French fried onions and parsley to taste. Dip into the au jus and enjoy!