Remove the stems from the portobello mushrooms. In a small bowl, mix the balsamic vinegar and olive oil. Place the mushrooms on a baking sheet. Brush the front and back of each mushroom with the marinade. Sprinkle with kosher salt and pepper, and let sit for 15 minutes.
Place the mushrooms on grill (or in a grill pan) and cook about 5 minutes per side, until juicy. In the last minute, place provolone cheese slices on top of each mushroom and grill until melted, then remove from the grill. Brush the ciabatta rolls with olive oil and place them on the grill until toasted.
Slice the tomato. Remove any tough stems from the arugula (not necessary for baby arugula).
Place each grilled portobello cap on half of a roll. Top with a spoonful of pesto, a tomato slice, a handful of arugula, and the other half of the roll.