Preheat oven to 350º. With an electric mixer, beat goat cheese, eggs, yolks, salt, and pepper together until smooth. Pour into tart shell. Bake 10–15 minutes, until set. (If making ahead, refrigerate tart; bring to room temperature before serving.)
Preheat oven to 400º. Drizzle beets with olive oil. Place beets in a baking dish and cover with foil. Bake 45 minutes or until fork can be inserted easily. Remove foil and peel beets while still warm; skins should slide right off. Quarter or dice.
Whisk together red wine vinegar, olive oil, salt, pepper, chives, and lemon balm. Pour over warm beets. Cool.
Just before serving, toss marinated beets with fresh arugula leaves and arrange on top of tart.