Dry brine the beef by seasoning it generously with salt and pepper. Place it in the fridge uncovered for at least 4 hours (or overnight).
Pull the beef out and let it sit at room temp for 30 minutes. Preheat oven to 325ºF.
Heat neutral oil in a large saucier or Dutch oven over medium-high heat. Sear the beef on all sides until well browned, then set it aside.
In the same pot, stir in tomato paste and cook until it darkens, about 1 minute.
Deglaze with sherry vinegar, scraping up all the fond from the bottom.
Return the beef to the pot. Add the stock, water, onion, celery, carrots, bell peppers, garlic, bay leaves, thyme, parsley, fennel seeds, coriander seeds, paprika, chili flakes, and peppercorns.
Bring to a simmer. Cover and transfer to the oven. Roast until the beef hits an internal temp of 125–130ºF, flipping halfway through, about 1–2 hours.
Let everything cool together. Transfer the beef and bell peppers to a container, strain the jus over them, and refrigerate overnight (or until the beef registers 40ºF). This rest is what makes the beef sliceable and deeply flavorful.
Gently warm the jus and peppers to about 140–160ºF.
Slice the beef thinly against the grain.
Cut French rolls into 6-inch lengths and split two-thirds of the way through to create a hinge. Wrap in foil and warm in the oven if desired.
Dip the sliced beef in the hot jus for about 30 seconds to warm it through.
Load the beef and peppers into the rolls. Dip the entire sandwich into the jus for 1–2 seconds if you want it wet.
Top with giardiniera-as much as you can fit. Wrap in parchment or foil and let sit 1 minute before serving.