Preheat oven to 425°F (220°C). Line a rimmed baking sheet with parchment paper.
On prepared pan, toss together peaches, brown sugar, and melted butter until peaches are well coated. Arrange peaches cut side down.
Bake until lightly browned on bottom, about 10 minutes. Turn peaches, and bake until fork-tender, 7 to 10 minutes more. Let cool completely on pan on a wire rack. Reserve 4 ounces (113 grams) roasted peaches. Chop remaining roasted peaches into small pieces. Reduce oven temperature to 325°F (170°C).
In the bowl of a stand mixer fitted with the paddle attachment, beat softened butter and granulated sugar at medium speed until light and fluffy, 3 to 4 minutes, stopping to scrape paddle and bottom and sides of bowl. Add eggs and egg white, one at a time, beating well after each addition.
In a large bowl, whisk together 2¾ cups (344 grams) flour, salt, baking powder, and five-spice powder. In a small bowl, stir together buttermilk and 1 teaspoon (4 grams) vanilla. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk mixture, beginning and ending with flour mixture, beating just until combined after each addition.
In a medium bowl, toss together chopped peaches and remaining 3 tablespoons (24 grams) flour to coat. Gently fold peach mixture into batter.
Spray a 10-cup Bundt pan with baking spray with flour. Spread batter into prepared pan. Tap pan on a kitchen towel-lined counter a few times to settle batter and release any air bubbles.
Bake until a wooden pick inserted near center comes out clean and an instant-read thermometer registers at least 205°F (96°C), 1 hour and 35 minutes to 1 hour and 45 minutes. Let cool in pan for 15 minutes. Invert cake onto a wire rack, and let cool completely.
Using a rubber spatula, press reserved roasted peaches through a fine-mesh sieve into a small bowl, reserving liquid.
In another small bowl, whisk together 3 tablespoons (45 grams) reserved peach liquid, confectioners’ sugar, and remaining ½ teaspoon (2 grams) vanilla until smooth and well combined. Pour onto cooled cake. Let stand until glaze is set before serving, about 30 minutes. Cover and refrigerate for up to 3 days.