Step 1: Season and sear the pot roast: Pat a 3 ½- to 4-pound beef chuck roast dry and season generously with kosher salt and black pepper. Heat a large Dutch oven over medium-high heat with a little neutral oil. Sear the roast on all sides until deeply browned. Don't rush this step. The browned exterior adds flavor to both the beef and the cooking liquid. Remove the roast and set it aside.
Step 2: Build the cooking liquid & slow cook: Add chopped onion and garlic to the same Dutch oven and cook until softened. Pour in beef broth and scrape up the browned bits from the bottom of the pot. Add Worcestershire sauce, Italian seasoning, and about ½ cup drained and chopped giardiniera. Return the roast to the pot. The liquid should come about one-third to halfway up the sides of the roast. Cover the Dutch oven and place it in a 325°F oven. Cook for about 3 to 3 ½ hours, or until the beef is very tender and easily pulls apart with a fork. The exact cooking time will depend on the size and shape of your roast. Tenderness is a better indicator than the clock. Transfer the beef to a cutting board and let it rest for about 10 minutes before chopping.
Step 3: Make the brown gravy: Strain the cooking liquid, reserving the liquid and discarding the solids. Skim excess fat from the top. In a saucepan, melt butter and whisk in flour. Cook for a minute or two, then gradually whisk in about 2 cups of the reserved beef cooking liquid. Simmer until the gravy thickens enough to coat the back of a spoon. Taste before adding additional salt. The giardiniera, broth, and Worcestershire sauce may have already brought plenty of seasoning.
Step 4: Make the giardiniera cheese sauce: Pour the evaporated milk into a medium saucepan and heat over medium-low heat until it is hot and just beginning to steam. You want it hot enough to melt the cheese but not boiling. Reduce the heat to low. Add the shredded Mexican cheese blend a small handful at a time, whisking or stirring constantly and allowing each addition to melt before adding more. Once the cheese sauce is smooth, stir in the garlic powder, onion powder, smoked paprika, and black pepper. Fold in the finely chopped, well-drained giardiniera. If the sauce is thicker than you'd like, add a small splash of warm evaporated milk or regular milk and stir until it reaches a smooth, pourable consistency. Keep the sauce warm over very low heat, stirring occasionally, until you're ready to assemble the nachos.
Step 5: Chop the pot roast: Remove any large pieces of fat or connective tissue from the pot roast and chop the beef into bite-size pieces. You don't want enormous chunks of beef on your nachos. Smaller pieces distribute the pot roast throughout the chips so you get some with nearly every bite.
Step 6: Build the nachos: Spread an even layer of chips over a platter or small plates. Scatter chopped pot roast over the chips. Drizzle generously with the giardiniera cheese sauce. Add a few spoonfuls of brown gravy, being careful not to completely soak the chips. Top with chopped tomatoes, sour cream, pickled jalapeños, scallions, and additional giardiniera if desired. Serve immediately while the chips are still crisp.