Special equipment: a deep fryer (optional) and a deep-fry thermometer
Heat a deep fryer filled with vegetable oil to 350 degrees F (or use a deep heavy-bottomed pot).
In a large bowl, whisk together the flour, baking powder and salt and set aside.
In a medium bowl, add the hot espresso and stir in the butter to melt. When melted, add the sugar and stir to combine. Add the milk and egg and whisk until the ingredients are well combined. Pour the wet ingredients into the dry ingredients and stir with a rubber spatula to form a batter, being careful not to overmix.
Using a small cookie scoop, scoop mounds of batter into the hot oil in batches, being careful not to splash. Fry, flipping often with a slotted spoon, until the outsides are deep golden brown, about 5 minutes. Transfer the donut bites to a plate lined with paper towels and let drain for 2 minutes. Roll the donut bites in sugar to coat. Serve warm.