Turn your pressure cooker to the saute setting. Wait until the pot is hot, then add the olive oil.
Season the oxtails with salt and black pepper. Place the largest pieces into the pot and sear them for 5 minutes per side until brown. Remove the meat and set it aside.
Add the sliced onion to the pot and cook for 2 minutes. Stir in the tomato paste, bay leaves, dried oregano, and dried thyme. Cook for 3 minutes, then add the chopped garlic and cook for 1 more minute.
Pour in the beef broth. Use a wooden spoon to scrape up any browned bits from the bottom of the pot.
Stir in the balsamic vinegar, soy sauce, and fish sauce. Return the browned oxtails to the pot.
Place the quartered potatoes and sliced carrots on top of the meat.
Secure the lid and set the valve to the sealing position. Cook on high pressure for 40 minutes. When the time is up, let the pressure release naturally for 20 minutes before opening the valve to release any remaining steam.
Carefully remove the oxtails and half of the carrots from the pot. Use a spoon to mash the potatoes and the remaining carrots directly into the cooking liquid to thicken the sauce.
Return the meat and reserved carrots to the thickened sauce. The beef is safe to eat when it reaches an internal temperature of 145°F (63°C) with a 3-minute rest, though this cooking method will take it much higher for tenderness. Serve warm.