Preheat oven to 350°F. Spray a 10-inch bundt pan generously with baking spray.
Combine the coffee and cocoa powder in a microwave-safe bowl. Microwave for 1 ½-2 minutes, whisking well every 30 seconds. Stir in the Kahlua, then set aside to cool.
¾ cups brewed coffee, ¾ cup cocoa powder, ½ cup Kahlua
In a stand mixer fitted with a paddle attachment or in a bowl with a hand mixer, combine the sugar, kosher salt, baking soda, eggs, and egg yolk on low speed for 1 minute. Scrape sides of bowl with rubber spatula.
2 ¼ cups sugar, 1 ¼ teaspoon kosher salt, 2 ½ teaspoon baking soda, 2 eggs + 1 egg yolk
Add buttermilk, oil, and vanilla and mix on low speed for 1 minute. Scrape sides of bowl.
1 ¼ cups buttermilk, 1 cup vegetable oil, 1 ½ teaspoon vanilla
Add flour and mix on medium for 2 minutes. Scrape sides of bowl.
2 ½ cups flour
Add cooled coffee-cocoa powder mixture and mix on medium speed for 3 minutes. Scrape sides of bowl.
Pour batter into prepared bundt pan. Bake on the center rack for 50-60 minutes, or until a toothpick inserted in the middle comes out clean.
Allow cake to cool for 15 minutes in the pan. Invert the warm cake on a cooling rack and allow to cool fully.