In a medium saucepan over medium heat, whisk together the milk and flour. Cook, whisking constantly, for 3 to 5 minutes, until the mixture thickens into a very thick pudding consistency and no lumps remain. It should coat the back of a spoon.

Remove from heat and transfer the mixture to a bowl. Let it cool completely to room temperature before continuing. (This step is important so the frosting doesn’t melt later.)

In a stand mixer fitted with the paddle attachment, cream the butter and sugar on medium-high speed for 2 to 3 minutes, until light and fluffy.

Add the cooled milk mixture and vanilla extract. Beat on medium-high speed until smooth and fully combined.

Switch to the whisk attachment.

Dissolve the espresso powder in 1 teaspoon hot water, then add it to the frosting. Whip for 5 to 7 minutes, until light, fluffy, and silky smooth.