Preheat oven to 350°F. Place fillo shells on rimmed baking sheet and bake for 3- 5 minutes, or until shells are crisped. Set aside to cool.
Meanwhile, heat butter and olive oil in a large skillet over medium heat. Add onions, cover and cook over medium heat, stirring, until softened, about 10 minutes. Reduce heat to medium-low. Season with salt and pepper and cook, stirring occasionally, until onions are caramelized and very soft, 40-50 minutes.
Stir in goat cheese and heavy cream, gently heat until well mixed. Divide mixture across the tart shells. Top each with a dollop of Armanino pesto. Enjoy warm or at room temperature.
