Preheat one side of barbecue to medium. On hot side, toast halved rolls on both sides, 2 to 3 min. Move rolls to side of barbecue with indirect heat and spread each with 2 tsp (10 mL) goat cheese. Top each with 3 slices of salami, 2 slices of tomato and 1 slice Swiss cheese. Close lid and cook for 5 to 7 min. or until Swiss cheese is lightly browned. Remove rolls to a platter and insert toothpick into each. Tent with foil and set aside.
Place a cast iron (or heavy) skillet on hot side of barbecue with indirect heat and melt butter. Working in 2 batches, crack in eggs, season with salt and pepper and cook with lid closed, 3 to 4 min. When egg whites are firm but yolks are still soft and shiny, gently remove fried eggs to a platter. Divide arugula among 8 rolls, place egg on top of each and serve.