Put tangerines in a large pot and cover with water. Toss in the vanilla bean - do not slice open! Bring to a boil, reduce heat to medium, and cook whole tangerines for about 2 hours until they are tender (soft when you squeeze them gently with tongs). Remove from pot and allow to cool. Scrape vanilla bean and set seeds aside.
Preheat oven to 375°F.
Over a large bowl, cut tangerines and remove any seeds. Squeeze to release excess water.
Line the bottom of an 8-inch springform pan with parchment paper. Spray the bottom and sides of the pan.
In a food processor, grind almonds until they turn into a course, powdery substance. Add the tangerines (with skin) and grind until the mixture forms a smooth, thick paste (some chunks of almonds and tangerines will be visible).
In a large bowl, beat eggs. Add sugar and vanilla bean seeds and mix. Add baking powder and salt.
Whisk tangerine-almond mixture into the egg mixture.
Pour batter into the pan. Bake 60-70 minutes, rotating halfway through baking, until toothpick comes out clean. Cool completely on rack.
Make sauce: Mix blueberries with powdered sugar and basil leaves in a small saucepan. Heat over medium heat, adding about 2 tablespoons of water. Cook until juices form. Sweeten with more sugar to taste. At this point, you may either puree the mixture or simply remove the basil leaves and leave it clumpy. Serve over cooled cake.