In a large saucepan, bring apple cider to a boil over medium-high heat. Boil 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. Spray a 10-cup Bundt pan with baking spray with flour.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, 1 cup (200 grams) granulated sugar, and vanilla at medium speed until creamy, 4 to 5 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition.

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. 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. 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 onto a wire rack, and let cool for 30 minutes.

In a small bowl, whisk together confectioners’ sugar and 1 tablespoon water until smooth. Brush cake with mixture, 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 over cake. Serve warm or at room temperature.