Preheat the oven to 425°F. Clear some space in the fridge—you’ll need to place a baking sheet with the puff pastry in there to chill while you work on the other components of the recipe.
Carefully unroll the thawed puff pastry and place it on a parchment-lined baking sheet. Whisk the egg in a small bowl. Brush the outer border of the pastry with the egg, saving the leftover for later. Fold the edges over to create a half-inch border all the way around, pressing to adhere. Prick small holes with a fork all over the surface of the pastry within the folded border. Place the baking sheet with the pastry in the fridge to chill while you continue with the recipe.
1 sheet puff pastry, 1 large egg
Trim off and discard the leafy green tops of the carrots if they are attached. Cut the carrots in half lengthwise.
1 lb. rainbow carrots
Heat 1 tablespoon of olive oil and the butter in a large skillet over medium heat. Add the halved carrots and season with a pinch or two of salt. Cook, flipping the carrots every minute or so, until they have softened. This will take 5 to 10 minutes, depending on their size and thickness. Add the honey, toss to coat, then remove the carrots from the heat. Set them aside to cool.
6 tbsps. olive oil, 1 tbsp. unsalted butter, sea salt, 1 tbsp. honey
Add the pistachios, parmesan, and garlic to a food processor. Pulse until finely chopped. Add the parsley, tarragon, lemon zest, and a pinch of salt. Pulse again until the herbs are finely chopped. Scrape down the sides.
1/3 cup raw pistachios, 1/4 cup grated Parmesan cheese, 1 large clove garlic, 1 cup flat leaf parsley (leaves only), 1/3 cup fresh tarragon, 1 lemon
With the processor running on low, stream in the remaining 5 tablespoons of olive oil until a thick pesto forms. Add the ricotta and pulse a few times to blend, scraping down the sides as needed.
1 cup whole milk ricotta cheese
Remove the pastry from the fridge. Spread the pesto-ricotta mixture in an even layer within the border. Arrange the roasted carrots over the surface, fitting them without overlapping.
Brush the remaining egg over the border of the crust again. Bake the tart in the preheated oven for 23 to 28 minutes, or until the pastry is puffed and golden. Slice and serve warm or at room temperature.