Preheat oven to 325°F. Melt 1⁄4 cup of the butter in a small saucepan over medium; stir in 1⁄2 cup of the brown sugar. Cook, whisking constantly, until sugar has dissolved and mixture is thoroughly combined, about 1 minute. Pour mixture evenly into a 15-cup (10 1⁄2-inch) Bundt pan that has been heavily greased with cooking spray. Place peach slices in an even layer on top of brown sugar mixture.

Beat cream cheese and remaining 1 1⁄2 cups butter with a heavy-duty stand mixer on medium speed until creamy, about 1 minute. Gradually add granulated sugar and remaining 1 1⁄2 cups brown sugar, beating on medium speed until light and fluffy, 3 to 5 minutes. Add eggs, 1 at a time, beating just until yolk disappears. Beat in 1 teaspoon of the vanilla.

Sift together flour and salt; add to butter mixture alternately with bourbon, beginning and ending with flour mixture. Spoon batter carefully over peaches in prepared pan; level with a spatula.

Bake in preheated oven until a long wooden pick inserted in center of cake comes out clean, 1 hour and 25 minutes to 1 hour and 30 minutes. Cool in pan on a wire rack until pan is cool enough to handle but still hot, about 25 minutes. Remove cake from pan to rack, and cool completely, about 2 hours.

Stir together powdered sugar, remaining 1⁄2 teaspoon vanilla, and 1 tablespoon of the milk in a small bowl, adding remaining 1 tablespoon milk, 1 teaspoon at a time, if needed to reach desired consistency. Drizzle over cooled cake.