Preheat the oven to 350°F. Grease a loaf pan and line it with parchment paper if desired.
In a large bowl, combine the melted butter, oil, and granulated sugar.
Add the eggs and vanilla extract, mixing until combined.
Stir the espresso powder into the warm milk. Add the coffee mixture and sour cream to the wet ingredients.
In a separate bowl, whisk together the flour, salt, baking soda, and baking powder.
Add the dry ingredients to the wet ingredients. Mix just until combined, taking care not to overmix.
Pour the batter into the prepared loaf pan and smooth the top.
Bake for 45–50 minutes, checking the center with a toothpick near the end of the baking time.
Allow the loaf to cool briefly in the pan, then transfer it to a wire rack and cool completely.
For the Coffee Buttercream
Beat the softened butter until smooth and creamy.
Gradually add the confectioners' sugar, mixing until incorporated.
Dissolve the instant coffee in a small amount of warm milk and allow it to cool completely.
Add the cooled coffee mixture gradually to the buttercream, beating until smooth and fluffy.
Spread the buttercream over the completely cooled Coffee Loaf. Slice and serve.
