Slice the mushroom caps into a 4 oz. portion. Brush the top side of the mushroom with 1 Tbsp. of olive oil. Place on a heated char-grill or broiler. Cook until just warmed through. Season with salt and pepper.
Brush one side of 2 slices of bread with 1 Tbsp. of olive oil each.
Place 2 slices of provolone on top of the bottom piece of bread with the oiled-side down. Spread 1 Tbsp. of tomato pesto on top of the cheese. Place ½ oz. of arugula on top of the pesto.
Slice the mushroom on a bias and place it on top of the arugula. Sprinkle 1 oz. of goat cheese on top of the mushroom. Place the second piece of bread, oiled-side up, on top of the sandwich.
Place the sandwich in a heated panini press, flat griddle or grill. Cook on both sides until golden brown and just warmed through.
Slice the sandwich in half on a bias. Place on a warmed serving plate.