In a food processor, combine flour, butter, baking powder, and 1/2 teaspoon salt. Pulse until mixture resembles coarse meal. While pulsing, gradually add ice water, until a dough begins to form. Turn onto a piece of plastic wrap, and pat into a 1-inch-thick square. Wrap, and refrigerate until firm, about 30 minutes.
Meanwhile, make filling: In a large skillet, heat oil over medium. Add onion, garlic, and carrot; cook, stirring occasionally, until carrot is tender, 5 to 7 minutes. Add turkey; cook, breaking up meat with a spoon, until no longer pink, 3 to 5 minutes. Add green beans and cream; cook until sauce is slightly thickened and green beans are tender, 2 to 4 minutes. Season with salt and pepper. Cool to room temperature, about 10 minutes.
Preheat oven to 400 degrees. On a lightly floured work surface, roll chilled dough into a 16-inch square; cut into 4 equal squares. Dividing evenly, place filling on one half of each square, leaving a 1/4-inch border around the filling. Brush border with water; fold dough over filling to enclose completely. Press edges to seal, then crimp with a fork. With the tip of a paring knife, cut 2 to 3 small vents on top. Transfer pies to two rimmed baking sheets; bake until golden brown, 25 to 30 minutes, rotating sheets halfway through.