Using a small paring knife, create four evenly spaced slits about 1/2 inch deep on the top of the roast. Insert a piece of garlic and a few rosemary leaves into each slit. Repeat on the opposite side of the roast. For the ends of the roast, make two additional slits and fill them with the remaining garlic and rosemary.

In a small bowl, mix 5 teaspoons of kosher salt with 1 tablespoon of ground black pepper. Coat the exterior of the roast with this mixture, pressing it firmly into the meat, and ensuring to fill the slits.

Place the trivet into the bottom of the 6-quart Instant Pot® insert. Add the beef broth and 1/4 cup of the red wine. Place the roast on the trivet, with the fat side facing upward. Set the Instant Pot® to pressure cook on low for 10 minutes, activating the "keep warm" feature (refer to the manufacturer’s guide for model-specific settings). Once the cooking cycle completes, allow the pot to rest, sealed, for 45 minutes. Afterward, open the lid and transfer the roast to a plate. Remove the trivet and pour the broth into a heatproof container. Rinse the insert and return it to the Instant Pot®.

Switch the Instant Pot® to sauté mode and add the butter. When the butter becomes foamy, sear the roast on all sides until a dark brown crust forms. The internal temperature of the roast should reach 120°F for medium-rare, requiring approximately 4 minutes per side. Move the roast to a cutting board and let it rest for 15 minutes before slicing.

For the gravy, add the remaining 1/4 cup of red wine to the Instant Pot®, scraping the browned bits from the bottom. Cook until most of the wine has evaporated, then sprinkle in the flour. Stir to create a smooth paste, then gradually whisk in the reserved broth. Bring the mixture to a boil and simmer for about 5 minutes, ensuring the raw flour taste is cooked out. If the gravy thickens too much, add extra beef broth. Season to taste, strain through a mesh sieve, and stir in the chopped herbs before serving. Present the gravy alongside the prime rib.