Since no one wants the soggy middle piece of a square tart, we cut a sheet of puff pastry in half to make two small ones: crisp edges for everyone! Eat as a meal with a salad or slice into petite wedges for a grab-and-go party app.
Preheat oven to 425°. Cut pastry in half. On parchment paper, roll each piece to a 6x12-inch rectangle.
In a large skillet, heat butter and oil over medium-high heat. When foaming, add carrots. Cook, stirring occasionally, until edges of carrots are lightly golden, about 5 minutes. Add vermouth and 1 tablespoon water; cover, reduce heat, and steam 1 minute. Season carrots with 1/4 teaspoon each kosher salt and pepper.
In a medium bowl, lightly beat one egg. Add ricotta, goat cheese, Parmesan, mint, lemon zest and the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper, stirring to combine.
Transfer parchment with pastry to a large baking sheet. Using a sharp knife, gently score a 1/2-inch border around each tart, being careful not to cut all the way through.
Separate the remaining egg into two small dishes. Lightly beat both the yolk and the white. Brush edges of tarts with beaten yolk. Prick inside of tarts with fork. Bake on bottom oven rack until edges are nicely golden brown, about 10 minutes. Remove from oven, but leave oven on and reduce temperature to 375°. Immediately press centers of tarts with a metal spatula to flatten. Cool 5 minutes. Brush centers of tarts evenly with egg white (to create a moisture barrier between the crust and the filling).
Divide cheese mixture between tarts and spread to the inside edges of borders. Arrange carrots evenly over filling. Bake tarts at 375° until fillings are set, about 20 minutes. Cool on a wire rack at least 15 minutes. Garnish with lemon zest and mint. Serve warm or at room temperature.