Combine coffee and milk in a small bowl or liquid measuring cup and set aside.
1/3 cup brewed coffee, 1/3 cup milk
In a large mixing bowl or the bowl of a stand mixer, beat together brown sugar and vegetable oil until well mixed. Add eggs and molasses and beat until smooth.
3/4 cup brown sugar, 2/3 cup vegetable oil, 2 large eggs, 1/2 cup molasses
Add 1/3 of the flour mixture to the wet ingredients and mix until completely incorporated. Scrape down the sides and bottom of the bowl. Pour in half of the coffee milk and mix until blended.
Repeat, then finish with the remaining flour mixture, mixing until smooth. Scrape the sides and bottom of the bowl, making sure everything is incorporated and there are no dry bits.
Pour batter into the prepared pan. Bake for 55-65 minutes, until a toothpick inserted into the center comes out clean. For the last 10 minutes of baking, you may want to tent with foil to prevent the top from browning too much while the center finishes baking.
Let cake cool in the pan for 10-15 minutes, then turn out onto a wire rack to cool completely. Make sure it has cooled before icing.
To make the icing, whisk together powdered sugar, milk, and vanilla until smooth and pourable, but not runny. Add more sugar or milk as needed, a little at a time. Pour evenly over the top of the cake. Let harden before serving.