Put the beet in a medium saucepan and add the balsamic vinegar, brown sugar, peppercorns, bay leaf, 3/4 cup cider vinegar, 1 tablespoon salt and 3 cups water and bring to a boil. Let boil until the sugar is completely dissolved, about 5 minutes. Add the peeled eggs to the pan, then cover and refrigerate for at least 4 hours and up to overnight for a deeper pink color.
Cut the eggs in half, then scoop out the yolks and place them in a medium bowl. Put the egg white halves on a serving platter. Add the mayonnaise, Dijon, chives, remaining tablespoon cider vinegar and a bunch of grinds of black pepper to the bowl with the yolks. Use a fork to mash the yolks until smooth and creamy and well combined. Transfer the mixture to a large resealable plastic bag and snip about 1/2-inch off of one of the corners. Pipe the mixture into the holes of the egg whites. Sprinkle the chopped pistachios over the top, then place 1 piece baby arugula on top of each egg. Sprinkle the platter with the crumbled goat cheese.