Make the Chai Latte Loaf
Preheat the oven to 350°F. Grease a standard 9 × 5-inch loaf pan and line it with parchment paper, leaving a little overhang along the sides for easy removal.
In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, combine the softened butter, granulated sugar, and dark brown sugar. Beat on medium-high speed for approximately 3–4 minutes, until light, fluffy, and slightly pale in color.
Add the eggs one at a time, mixing well after each addition. Add the additional egg yolk and beat until incorporated. Scrape down the sides and bottom of the bowl as needed.
Mix in the sour cream, cooled espresso or coffee, and vanilla bean paste until combined. The mixture may appear slightly curdled at this stage, which is normal.
In a separate medium bowl, whisk together the all-purpose flour, cornstarch, chai spice, espresso powder (if using), baking powder, baking soda, and kosher salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed just until combined. Scrape down the sides of the bowl as needed, being careful not to overmix.
Spoon the batter into the prepared loaf pan and spread it into an even layer using an offset spatula.
Bake at 350°F for approximately 45–55 minutes, or until the center is set and a toothpick inserted into the middle comes out with a few moist crumbs but no wet batter. If the top begins browning too quickly, loosely tent the loaf with aluminum foil.
Allow the loaf to cool in the pan for approximately 15 minutes. Carefully lift it out using the parchment paper and transfer it to a wire rack to cool completely before icing.
Make the Espresso Icing
In a small mixing bowl, whisk together the sifted powdered sugar, 1 tablespoon cooled espresso or coffee, vanilla bean paste, and a pinch of kosher salt.
Whisk until smooth. If the icing is too thick, add additional espresso a few drops at a time until it reaches a thick but drizzleable consistency.
Once the loaf has cooled completely, drizzle the espresso icing generously over the top using a spoon or small whisk.
Let the icing set for several minutes before slicing and serving.