Preheat oven to 350°F (175°C). Grease a 9x5-inch loaf pan and line with parchment. Whisk flour, baking powder, baking soda, and salt in a bowl.

Dissolve 2 tbsp instant coffee in 2 tbsp hot water and let cool. Beat butter and sugar until fluffy, about 2 minutes. Add eggs one at a time, then vanilla and cooled coffee mixture.

Mix in one third of the flour mixture on low, then add all the sour cream, another third of flour, and finally fold in the rest by hand with a spatula.

Scrape batter into pan; bake 50-55 minutes until golden and a toothpick comes out with moist crumbs. Cool in pan 10 minutes, then lift out and cool completely on a rack.

Whisk powdered sugar, the second dissolved coffee mixture, and 2 tbsp milk. Drizzle over cooled loaf, let set 5 minutes, and slice.