Gently toss together the chuck and sirloin in a large bowl, making sure not to overwork the mixture.
Combine the dried parsley, fennel seed, granulated garlic and red pepper flakes in a small bowl. Sprinkle the spice mixture over the beef along, with the Parmesan, and fold gently to combine. Form the beef into four 3/4-inch-thick patties, each about 4 inches in diameter. Sprinkle each generously with salt and a few grinds of black pepper.
Place the patties in the hot skillet; do not press down with a spatula. When the patties are seared and browned on one side, about 3 1/2 minutes, flip and cook on the other side until browned and the burgers are medium-rare, about 3 1/2 minutes.
Split the hamburger buns. Arrange some arugula on the bottom buns and top each with a slice of tomato, a burger patty and a dollop of pesto. Top with the top halves of the rolls Serve with giardiniera.
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