Preheat oven to 325°F (170°C). Spray a 10-cup Bundt pan with baking spray with flour.
In a small saucepan, melt 2 tablespoons (28 grams) butter over medium-high heat. Add brown sugar and lemon juice, and cook, stirring occasionally, until sugar is melted and mixture is combined. Pour into prepared pan, spreading with a spatula as needed. Place peach slices on top of caramel.
In the bowl of a stand mixer fitted with the paddle attachment, beat granulated sugar and remaining 1 cup (227 grams) butter at medium speed until light and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition.
In a medium bowl, whisk together 2½ cups (313 grams) flour, salt, and baking powder. In a small bowl, stir together 1 cup (240 grams) milk and 1½ teaspoons (9 grams) vanilla bean paste. With mixer on low speed, gradually add flour mixture to butter mixture alternately with milk mixture, beginning and ending with flour mixture, beating just until combined after each addition.
In another medium bowl, toss together chopped peaches and remaining 2 tablespoons (16 grams) flour to coat. Gently fold peaches and Cobbler Crumbs into batter. Spoon batter into prepared pan, leveling as needed. Firmly tap pan on a kitchen towel-lined counter 7 to 10 times to level 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), about 1½ hours. Let cool in pan for 20 minutes. Invert cake onto a wire rack, and let cool completely. (Some peach slices may remain in the pan. Use an offset spatula to lift and place onto cake.)
In another small bowl, whisk together confectioners’ sugar, 1 tablespoon (15 grams) milk, and remaining ½ teaspoon (3 grams) vanilla bean paste until combined; add up to remaining 1 tablespoon (15 grams) milk, if needed. Drizzle onto cooled cake. Refrigerate in an airtight container for up to 3 days.