Combine the ground chuck, 1 1/2 teaspoons salt, and pepper in a medium bowl. Shape the mixture into 4 (1/2-inch-thick) patties.
Slice the tomatoes into ½ inch rounds and place on a baking tray. Season both sides with the paprika and remaining salt. Rub with the olive oil to coat.
Prepare the barbecue (medium-high heat). Grill the burgers until cooked to desired doneness, about 4 minutes per side for medium-rare. On a clean, oiled area of the grill, grill the tomatoes until just charred, about 3 minutes per side. Grill the ciabatta squares cut side down, until lightly toasted, about 2 minutes.
Place the burgers on the bottom half of the ciabatta. Top each burger with 2 tablespoons gorgonzola and a grilled tomato slice. Arrange a handful of arugula on each. Cover the burger with the roll tops and serve.