Line a rimmed baking sheet with parchment paper.

On a lightly floured surface, roll All-Butter Piecrust into a 10½-inch square (about ¼ inch thick). Using a sharp knife, trim ¼ inch from all sides of dough. Using a bench scraper or the back of a knife, lightly score a square into dough 1 inch from edges. Carefully transfer dough to prepared pan.

In the work bowl of a food processor, pulse feta, ricotta, chives, 1 tablespoon oil, basil, garlic powder, ½ teaspoon pepper, and thyme until smooth and creamy, 1 to 2 minutes. Spread mixture onto dough within scored square. Freeze for 15 minutes.

Position oven rack in bottom third of oven. Preheat oven to 400°F (200°C).

Peel and halve carrots lengthwise, reserving tops for chimichurri. On a baking sheet, toss together carrots, salt, remaining 1 tablespoon oil, and remaining ½ teaspoon pepper. Arrange carrots on feta mixture.

Bake for 20 minutes. Cover with foil, and bake until crust is golden brown, 10 to 15 minutes more. Let cool on pan for 15 minutes. Top with Carrot Top Chimichurri. Best served warm. Refrigerate in an airtight container for up to 2 days.

In a large bowl, whisk together flour and salt. Add cold butter, and toss with your hands until butter is coated in flour. Using a pastry blender or your hands, work butter into flour mixture until butter pieces are quarter-size. Add ¼ cup (60 grams) ice water, and toss until flour is hydrated and a dough begins to form. (Mixture may appear crumbly. It should be moist and hold together when pinched.) Add more ice water, 1 tablespoon (15 grams) at a time as needed, until dough comes together when pressed. (Be careful not to add too much water.)

Turn out dough, and press into a flat square. Wrap tightly in plastic wrap, and refrigerate for at least 30 minutes or up to 2 days.

Turn out dough onto a lightly floured surface, and roll into a 10-inch square. Fold dough in half; fold in half again. (This mimics lamination.) Wrap in plastic wrap, and refrigerate for at least 30 minutes or up to 3 days, or freeze for up to 6 months. Let frozen dough thaw overnight in refrigerator before using.

In a small bowl, stir together all ingredients until well combined. Refrigerate in an airtight container for up to 1 week.