Combine the ground chuck, 1 1/2 teaspoons of the salt, and the pepper in a medium bowl and shape into six 1/2-inch-thick patties.
Slice the tomatoes into 1/3-inch-thick rounds and place on a baking sheet. Sprinkle with the paprika and the remaining 1/2 teaspoon salt. Rub with the olive oil.
Preheat a grill or grill pan to medium-high heat. Grill the burgers until cooked to the desired doneness, about 3 minutes per side for medium-rare. On a clean, well-oiled area of the grill, grill the tomatoes until just charred, about 2 minutes per side. Grill the ciabatta rolls, cut-side down, until lightly toasted, about 2 minutes.
Place the burgers on the bottom halves of the rolls. Top each with 1 to 2 tablespoons of the gorgonzola, a tomato slice and some arugula. Cover with the roll tops and serve.