In a large saucepan, bring cider to a boil over medium-high heat. Cook until cider is reduced to 1½ cups, about 1½ hours, stirring frequently during last 30 minutes. Remove from heat; let cool completely.
Preheat oven to 325°F (170°C).
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and 1 cup (200 grams) granulated sugar at medium speed until creamy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition. Beat in vanilla.
In a medium bowl, whisk together flour, 1 tablespoon (6 grams) cinnamon, salt, baking powder, and baking soda. In a small bowl, stir together reduced cider, milk, and vinegar. With mixer on low speed, gradually add flour mixture to butter mixture alternately with cider mixture, beginning and ending with flour mixture, beating just until combined after each addition. Stir in apple.
Spray a 10-cup Bundt pan with baking spray with flour. Pour batter into prepared pan.
Bake until a wooden pick inserted near center comes out with a few moist crumbs, about 1 hour. Let cool in pan for 10 minutes. Invert cake onto a wire rack, and let cool for 30 minutes.
In a small bowl, whisk together confectioners’ sugar and 1 tablespoon (15 grams) water until smooth. Brush mixture onto cake, working quickly to keep moist.
In another small bowl, combine remaining ¼ cup (50 grams) granulated sugar and remaining 1 tablespoon (6 grams) cinnamon; sprinkle onto cake. Serve warm or at room temperature.