Preheat Panini grill or grill pan. Take 4 slices of the bread and line them up on a sheet pan. Brush one side of each slice with olive oil. Place the slices olive oil side down on the sheet pan. On 4 of the bread slices, spread with some mustard. Layer with Brie and the nectarine. Make sure to layer each slice of bread evenly. Drizzle with honey, top with arugula. Season with salt and pepper.
Place the other 4 bread slices on top to make 4 sandwiches. Brush more oil on each of the 4 top bread slices. Place the sandwiches, 2 at a time, on the Panini grill. If you’re using a grill pan, place the sandwiches in the grill pan (2 at a time) and place a heavier pan on top of the sandwiches to press them down.
Cook until the sandwiches are toasty golden brown and the cheese is ooey gooey and melted, about 3-5 minutes.