How to Make It
Preheat oven to 450°F. Heat 2 teaspoons of the oil in a large skillet over medium-high. Add onion, and cook, stirring occasionally, until lightly browned, about 3 minutes. Reduce heat to medium-low, and continue to cook, stirring occasionally, until caramelized, 15 to 20 minutes. Transfer to a medium bowl to cool completely, about 15 minutes.
Meanwhile, toss together carrots, 1 tablespoon of the oil, 1/4 teaspoon of the salt, and 1/8 teaspoon of the pepper. Arrange in a single layer on a rimmed baking sheet. Bake in preheated oven until just tender and lightly charred, about 15 minutes
Place 1 phyllo sheet on a baking sheet lined with parchment paper. (Cover remaining phyllo sheets with a damp towel to keep them from drying out.) Lightly brush phyllo sheet with some of the oil, and top with another phyllo sheet. Repeat layers with remaining phyllo sheets and oil, reserving 1 teaspoon oil to brush on carrots.
Remove carrots from oven, and reduce oven temperature to 400°F. Add ricotta, chives, tarragon, garlic, 1/2 teaspoon of the salt, and remaining 1/8 teaspoon pepper to caramelized onions; stir to combine. Spread ricotta mixture on phyllo stack, leaving a 1/2-inch border. Arrange carrots, cut side up and side-by-side, over ricotta mixture. Brush carrots with remaining 1 teaspoon oil. Bake at 400°F until phyllo is browned and crisp, about 20 minutes. Sprinkle with remaining 1/4 teaspoon salt; top with shaved cheese, hazelnuts, and tarragon leaves.