Pat lamb shanks dry with paper towels; sprinkle with ½ teaspoon salt.
Set Instant Pot to high sauté function, add oil to pot and heat until smoking, about 5 minutes. Brown 2 shanks on all sides, 8–10 minutes. Transfer shanks to a plate; repeat with remaining shanks.
Add onion, carrots, and celery to pot; cook until vegetables start to soften, about 5 minutes.
Stir in garlic, tomato paste, herbs de Provence, and remaining ¼ teaspoon salt and cook 1 minute.
Stir in wine, scraping up any browned bits.
Nestle shanks into pot; adding any accumulated juices. Lock lid in place and close pressure release valve. Select high pressure cook functions and cook for 60 minutes.
Turn off Instant Pot and let pressure release naturally for 15 minutes. Quick-release any remaining pressure, then carefully remove lid, allowing steam to escape away from you. Transfer shanks to a serving dish, tent with foil, and let rest while preparing the sauce.
Skim fat off top of braising liquid in pot. Transfer de-fatted braising liquid and vegetables to blender, add vinegar and blend until smooth; season with salt and pepper taste.
Serve lamb shanks over mashed potatoes and spoon puréed sauce over top.