Pat pork dry using paper towels and salt lightly on all sides.
Heat oil in a 4- to 6-quart pressure cooker over high heat until smoking. Add pork; cook until browned, about 4 minutes per side. Add carrots, onion, and garlic; cook and stir for 1 minute. Pour in chicken broth, vinegar, honey, soy sauce, ginger, tomato paste, and black pepper. Bring sauce to a boil, scraping the browned bits from the bottom of the pan with a wooden spoon.
Close cooker securely and place pressure regulator over vent according to manufacturer's instructions. Heat until steam escapes in a steady flow, about 5 minutes. Cook at low pressure, 20 to 25 minutes. Let pressure release naturally according to manufacturer's instructions, 5 to 10 minutes. Unlock and remove lid.
Mix cold water and cornstarch together in a small bowl.
Remove pork to a warm platter and cover. Continue cooking sauce until reduced, 3 to 5 minutes. Stir in the cornstarch mixture; bring to a boil and cook for 1 minute. Add butter and stir gently to melt.
Slice pork across the grain; serve with the sauce.
