Heat a grill pan to medium-high heat.
Add the mushrooms and red peppers to a sheet tray, drizzle with olive oil and sprinkle with salt and pepper. Toss to coat. Grill the portobellos for 3 minutes per side, 6 minutes total. Grill the red peppers for 5 minutes per side, 10 minutes total. Remove from the grill and set aside until cool enough to handle.
Cut two 4-inch pieces of bread, then slice in half lengthwise. Spread the Herbed Cream Cheese Spread on each piece of bread. Pile on the grilled portobellos, red peppers and arugula on 2 slices of the bread and close the sandwiches.
Place the sandwiches on top of the hot grill pan and place a cast-iron skillet on top to press. Cook for 3 to 4 minutes, and then flip and cook 3 to 4 more minutes on the remaining side. Slice on a diagonal and serve.
Add the cream cheese, basil, chives and garlic to a bowl of a food processor and pulse until smooth and combined. Season with salt and pepper.